Lorentzian Geodesic Flows and Interpolation between Hypersurfaces in Euclidean Spaces
James Damon



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Lorentzian Geodesic Flows]
Lorentzian Geodesic Flows and Interpolation between Hypersurfaces in Euclidean Spaces
J. Damon]James Damon 
Partially supported by DARPA grant HR0011-05-1-0057 and 
National Science Foundation grant DMS-1105470
Department of Mathematics 

University of North Carolina 

Chapel Hill, NC 27599-3250  

USA








We consider geodesic flows between hypersurfaces in ^n.  However, rather than consider using geodesics in ^n, which are straight lines, we consider an induced flow using geodesics between the tangent spaces of the hypersurfaces viewed as affine hyperplanes.  For naturality, we want the geodesic flow to be invariant under rigid 
transformations and homotheties.  Consequently, we do not use the dual projective space, as the geodesic flow in this space is not preserved under translations.  Instead we give an alternate approach using a Lorentzian space, which is semi-Riemannian with a metric of index 1. For this space for points corresponding to affine hyperplanes in ^n, we give a formula for the geodesic between two such points.  As a consequence, we show the geodesic flow is preserved by rigid transformations and homotheties of ^n.  Furthermore, we give a criterion that a vector field in a smoothly varying family of hyperplanes along a curve yields a Lorentzian parallel vector field for the corresponding curve in the Lorentzian space.  As a result this provides a method to extend an orthogonal frame in one affine hyperplane to a smoothly Lorentzian varyingfamily of 
orthogonal frames in a family of affine hyperplanes along a smooth curve, as well as a interpolating between two such frames with a smooth minimally Lorentzian varyingfamily of orthogonal frames. We further give sufficient conditions that the Lorentzian flow from a hypersurface is nonsingular and that the resulting corresponding flow in ^n is nonsingular.  This is illustrated for surfaces in ^3. 


 
Primary: 53B30, 53D25, Secondary: 53C44

[
    [
    
=====



PRELIMINARY VERSION





§ INTRODUCTION
 


We consider the problem of constructing a natural diffeomorphic flow 
between hypersurfaces M_0 and M_1 of ^n which is in some sense 
both naturaland geodesicviewed in some 
appropriate space (as in figure   ).  
 
 There are several approaches to this question.  One is from the 
perspective  
of a Riemannian metric on the group of diffeomorphisms of ^n.  If the 
smooth hypersurfaces M_i bound compact regions _i , then the 
group of diffeomorphisms (^n) acts on such regions _i  and 
their boundaries.  Then,  if φ_t, 1≤ t ≤ 1, is a geodesic in 
(^n) beginning at the identity, then φ_t() (or 
φ_t(M_i)) provides a path interpolating between _0 = 
φ_0() = and _1 = φ_1().  Then, the geodesic 
equations can be computed and numerically solved to construct the flow 
φ_t.  This is the method developed by Younes, Trouve, Glaunes 
<cit.>, <cit.>, <cit.>, <cit.>, and Mumford, Michor 
<cit.>, <cit.> etc.  An alternate approach which we consider in this paper requires that we 
are given a correspondence between M_0 and M_1, defined by a 
diffeomorphism χ : M_0 → M_1, which need not be the restriction of 
a global diffeomorphism of ^n (and the M_i may have boundaries).  
Then, if we map M_0 and M_1 to submanifolds of a natural ambient 
space , we can seek a geodesic flowbetween M_0 and 
M_1, viewed as submanifolds of , sending x to φ(x) along 
a geodesic.  Then, we use this geodesic flow to define a flow between 
M_0 and M_1 back in ^n.  The simplest example of this is the radial flowfrom M_0 
using the vector field U on M_0 defined by U(x) = φ(x) - x.  
Then, the radial flow is the geodesic flow in ^n defined by 
φ_t(x) = x + t· U(x).  The analysis of the nonsingularity of the 
radial flow is given in <cit.> in the more general context of 
skeletal structures.  
This includes the case where M_1 is a generalized offset 
surfaceof M_0 via the generalized offset vector field U.   












In this paper, we give an alternate approach to interpolation via a geodesic 
flow between hypersurfaces with a given correspondence.  While the radial 
flow views each hypersurface as a collection of points, we will instead view 
each as defined by their collection of tangent spaces.  This leads to consideration 
of geodesic flows between dual varieties.  The dual 
varieties traditionally lie in the dual projective space.  However, 
 the geodesic flow induced on the dual projective space with its natural 
Riemannian metric does not have certain natural properties that are desirable, 
such as invariance under translation.  Instead, we shall define in <ref> a Lorentzian mapto a hypersurface 
^n in the Lorentzian space ^n+1 defined by their tangent 
spaces as affine hyperplanes in ^n+1.  So instead of representing 
hypersurfaces in terms of dual varieties, we instead 
represent them as subspaces of ^n+1, which is the subspace of 
points in Minkowski space ^n+2, 1 of Lorentzian norm 1.  Then, we 
use the geodesic flow for the Lorentzian metric on ^n+1, and then 
transform that geodesic flow back to a flow between the original 
manifolds in ^n.  To do this we determine in <ref> the explicit form for the 
Lorentzian geodesics in ^n+1 between points in ^n.  We show 
these geodesics lie in ^n and show in <ref>  that 
these geodesics are invariant under the extended Poincare group.  Given a 
Lorentzian geodesic between two points in ^n, there corresponds a 
smooth family of hyperplanes Π_t.  We further give in <ref> a criterion for Lorentzian parallel 
vector fields  in a family of hyperplanes Π_t along a curve 
≫(t) in ^n, and then determine the Lorentzian parallel vector fields over 
a Lorentzian geodesic corresponding to vector fields with values in Π_t.  Using 
this, we determine for an orthonormal frame {e_i, 0} in Π_0,  a 
smooth family of orthonormal frames {e_i, t} in Π_t which 
correspond to a Lorentzian parallel family of frames along the Lorentzian 
geodesic.  Using this we further determine a method for interpolating 
between orthonormal frames {e_i, 0} in Π_0 and {e_i, 1} in 
Π_1.  In <ref> and <ref> we relate the properties 
of hypersurfaces M̃ of ^n with corresponding properties of 
the envelopes formed from the planes defined by M̃.  In <ref> we give a diffeomorphism between the Lorentzian space 
^n with the dual projective space P^n ∨, which is a 
Riemannian manifold.  The classification of generic Legendrian 
singularities in P^n ∨ gives the form of the singular points of 
images and this is used to give criteria for the lifting to a hypersurface in 
^n+1 as the envelope of the family of corresponding hyperplanes.  
Finally, in section <ref>  we give in Theorem <ref> the 
existence and continuous dependence of the corresponding Lorentzian geodesic flow  between two hypersurfaces M_0 and 
M_1 in ^n+1  and in Theorem <ref> we give a sufficient 
condition for the flow to be nonsingular.  As a special case we consider in 
<ref> the results for surfaces in ^3. 


 

CONTENTS
 




  * Overview

 
  * Semi-Riemannian Manifolds and Lorentzian Manifolds 

 

  * Definition of the Lorentzian Map

 

  * Lorentzian Geodesic Flow on ^n+1

 

  * Invariance of Lorentzian Geodesic Flow and Special Cases 



  * Families of Lorentzian Parallel Frames on Lorentzian Geodesic 
Flows



  * Dual Varieties and Singular Lorentzian Manifolds



  * Sufficient Condition for Smoothness of Envelopes



  * Induced Geodesic Flow between Hypersurfaces



  * Results for the Case of Surfaces in ^3 

  











 

§ OVERVIEW

 
As mentioned in the introduction there are two main methods for 
deforming 
one given hypersurface M_0 ⊂^n to another M_1.  One is to 
find a path ψ_t in G, which is some specified a group of 
diffeomorphisms of ^n, from the identity so that ψ_1(M_0) = M_1 
(and ψ_0(M_0) = M_0).  Another approach involves constructing a geometric flow between M_0 
and 
M_1.  Several flows such as curvature flows do not provide a flow to a 
specific hypersurface such as M_1.  An alternate approach which we 
shall 
use will assume that we have a correspondence given by a diffeomorphism 
χ : M_0 → M_1 and construct a geodesic flowwhich at 
time t = 1 gives χ.  The geodesic flow will be defined using an 
associated space .  We shall consider natural maps 
φ_i : M_i → , where  is a distinguished space which 
reflects certain geometric properties of the M_i. 

    M_0      φ_0⟶   
    χ↓      φ_1↗   
    M_1






 

Given smooth maps φ_i : M_i → and a diffeomorphism χ : 
M_0 → M_1  A geodesic flow between the maps φ_i is a 
smooth map ψ̃_t :  M_0 × [0, 1] → such that for any 
x 
∈ M_0, ψ̃_t(x) : [0, 1] → is a geodesic from 
φ_0(x) to φ_1 ∘χ (x) 
 
We shall also refer to the geodesic flow as being between the M̃_i 
= 
φ_i(M_i). However, we note that it is possible for more than one 
x_i 
∈ M_0 to map to the same point in y ∈, however, the geodesic 
flow 
from y can differ for each point x_i.  

Then, we will complement this with a method for finding the 
corresponding 
flow ψ_t between M_0 and M_1 such that φ_t ∘ψ_t 
= 
ψ̃_t, where φ_t: ψ_t(M_0) →ψ̃_t(M_0).  We 
furthermore want this flow to satisfy certain properties.  A main property 
is that the flow construction is invariant under the action of the extended 
Poincare group formed from rigid transformations and homotheties (scalar 
multiplication).  
By this we mean: if M_0^' = A(M_0) and M_1^' = A(M_1) 
are transforms of M_0 and M_1 by a transformation A formed from 
the composition of a rigid transformation and homothety, and M_t is the 
flow between M_0 and M_1, then A(M_t) gives 
the flow between M_0^' and M_1^'.  We are specifically interested in a geodesic flowwhich will 
be a flow defined using the tangent bundles TM_0 to TM_1 so that we 
specifically control the flow of the tangent spaces.  At first, an apparent 
natural choice is the dual projective space P^n ∨.  Via the 
tangent bundle of a hypersurface M ⊂^n  (⊂ P^n) there 
is the natural map : M → P^n ∨, sending x ↦ T_xM.   
The natural 
Riemannian structure on the real projective space P^n ∨ is 
induced from S^n via the natural covering map S^n → P^n, so that 
geodesics of S^n map to geodesics on P^n ∨.  However, simple 
examples show that the induced geodesic slow on P^n ∨ is not 
invariant under translation in ^n.  For example, this Riemannian 
geodesic flow between the hyperplanes given by · = c_0 and 
· = c_1 is given by · = c_t,  where c_t = 
tan (tarctan (c_1) + (1-t)arctan (c_0)).  It is easily seen that if we 
translate the two planes by adding a fixed amount d to each c_i, then 
the corresponding formula does not give the translation of the first.  We will use an alternate space for , namely, the Lorentzian space 
^n+1 which is a Lorentzian subspace of Minkowski space 
^n+2, 1.   In fact the images will be in an n-dimensional 
submanifold ^n ⊂^n+1.  On ^n+1 it is classical that 
the geodesics are intersections with planes through the origin in 
^n+2, 1.  This allows a simple description of the geodesic flow on 
^n+1.  We transfer this flow to a flow on ^n using an inverse 
envelope construction, which reduces to solving systems of linear 
equations.  We will give conditions for the smoothness of the inverse 
construction which uses knowledge of the generic Legendrian 
singularities.  We shall furthermore see that the construction is invariant under the 
action of rigid transformations and homotheties.   In addition, uniform 
translations and homotheties will be geodesic flows, and a pseudo 
rotationwhich is a variant of uniform rotation is also a geodesic 
flow.  







§ SEMI-RIEMANNIAN MANIFOLDS AND LORENTZIAN MANIFOLDS

 
A Semi-Riemannian manifold M is a smooth manifold M, with a  
nondegenerate bilinear form < ·, · >_x on the tangent space 
T_xM, for eaxh x ∈ M which smoothly varies with x.  We do not 
require that < ·, · >_x be positive definite.  We denote the index 
of < ·, · >_(x) by ν.  In the case that ν = 1, M is 
referred to as a Lorentzian manifold.  A basic example is Minkowski space which (for our purposes) is ^n+2 
with bilinear form defined for v = (v_1, … , v_n+2) and  
w = (w_1, … , v_n+2)

    < v, w >_L     =   ∑_i = 1^n+1 v_i· w_i -   
    v_n+2· w_n+2

There are a number of different notations for this Minkowski space.  We shall 
use ^n+2, 1.  We shall also use the notation <·, · >_L for the 
Lorentzian inner product on ^n+2, 1.  A submanifold N of a semi-Riemannian manifold M is a 
semi-Riemannian submanifold if for each x ∈ N, the restriction of 
< ·, · >_(x) to T_xN is nondegenerate.  
There are several important submanifolds of ^n+2, 1.  One such is 
the Lorentzian submanifold

    ^n+1    =   { (v_1, … , v_n+2) ∈^n+2, 1 : ∑_i = 
    1^n+1 v_i^2   -   v_n+2^2   =  1},
  
which is called de Sitter space (see Fig. <ref>). 
A second important one is hyperbolic space ^n+1 defined by

    ^n+1    =   { (v_1, … , v_n+2) ∈^n+2, 1 : 
    ∑_i = 1^n+1 v_i^2   -   v_n+2^2    =    -1     
    v_n+2 > 0}.

 
 

By contrast the restriction of < ·, · >_L to ^n+1 is a 
Riemannian metric of constant negative curvature -1.  There is natural 
duality between codimension 1 submanifolds of ^ n+1 obtained as the 
intersection of ^ n+1 with a time-likehyperplane Π 
through 0 (containing a time-likevector z with <z, z>_L 
  < 0) paired with the points ± z^'∈^n+1 given where 
z^' lies on a line through the origin which is the Lorentzian 
orthogonal complement to Π. Many of the results which hold for Riemannian manifolds also hold for a 
Semi-
Riemannian manifold M.  
[Basic properties of Semi-Riemannian Manifolds (see 
<cit.>] 
 

For a Semi-Riemannian manifold M, there are the following properties 
analogous to those for Riemannian manifolds: 

  * Smooth Curves on M have lengths defined using 
| < ·, · > |.  

  * There is a unique connection which satisfies the usual properties 
of a Riemannian Levi-Civita connection.

  * Geodesics are defined locally from any point x ∈ M and with any 
initial velocity v ∈ T_xM.  They are critical curves for the length 
functional, and they have constant speed.

  * If N is a semi-Riemannian submanifold of M, then a constant 
speed curve ≫(t) in N is a geodesic in N if the acceleration 
≫^''(t) is normal to N (with respect to the 
semi-Riemannian metric) at all points of ≫(t).

  * Any point x ∈ M has a convex neighborhood  W, 
which has the property that any two points in W are joined by a unique 
geodesic in the neighborhood.

  * If ≫(t) is a geodesic joining x_0 = ≫(0) and x_1 = ≫(1) 
and x_0 and x_1 are not conjugate along ≫(t), then given a 
neighborhood W of ≫(t), there are neighborhoods of W_0 of x_0 
and W_1 of x_1 so that if x_0^'∈ W_0, and x_1^'∈ W_1, there is a unique geodesic in the neighborhood W from 
x_0^' to x_1^'.





Then, as an example, it is straightforward to verify that for any 
z ∈^n+1, the vector z is orthogonal to ^n+1 at the point 
z.  Suppose P is a plane in ^n+2, 1 containing the origin.  Let 
≫(t) be a constant Lorentzian speed parametrization of the curve obtained 
by intersecting P with ^n+1.  Then, by a standard argument similar 
to that for the case of a Euclidean sphere, ≫(t) is a geodesic. 

All geodesics of ^n+1 are obtained in this way.  It follows that the 
submanifolds of ^n+1 obtained by intersecting ^n+1 with a 
linear subspace is a totally geodesic submanifold of ^n+1.  


§ DEFINITION OF THE LORENTZIAN MAP

 
We begin by giving a geometric definition of a Lorentzian map from a 
smooth hypersurface M ∈^n, as a natural map from M to 
^n+1; and then giving that geometric definition an algebraic form.  


 §.§ Geometric Definition of the Lorentzian Map
 
First, we let S^n denote the unit sphere in 
^n+1 centered at the origin, and we let _n+1 = (0, …, 0, 1)  
∈^n+1.  Then, stereographic projection defines a map p: 
S^n\{_n+1}→^n sending y to the point where 
the line from _n+1 to y intersects ^n.  Given a hyperplane 
Π in ^n, it together with _n+1 spans a hyperplane 
Π^' in R^n+1.  We can identify R^n+1 with R^n+1×{_n+2}⊂ R^n+2, 1 by translation in the direction  
_n+2.  The intersection of this plane with S^n+1 is an 
n-sphere.  Then, via this identification of R^n+1 with the 
hyperplane in ^n+2, 1 defined by x_n+2 = 1, we form the 
hyperplane Π^'' in ^n+2, 1 spanned by 
Π^' together with 0.  This hyperplane is time-like because 
Π^'' intersects R^n+1×{_n+2} in a 
hyperplane Π^' which intersects the unit sphere in R^n+1×{_n+2} in a sphere, hence it intersects the interior disk.  
Then, the duality defined by the Lorentzian inner product associates to the 
hyperplane Π^'' the Lorentzian orthogonal line ℓ 
through the origin.  As the hyperplane is time-like, ℓ has non-empty 
intersection with ^n+1 in a pair of points z and -z.  
In order to obtain a single valued map, there are two possibilities:  Either 
we consider the induce map to ^n+1 =  ^n+1/∼, where 
∼ identifies each pair of points z and -z  of ^n+1; or we 
need on Π a unit vector field  orienting Π.  Given 
the normal vector , it defines a distinguished side of Π.  Then we 
obtain a distinguished side for Π^' and then 
Π^'', which singles out one of the two points in 
^n+1 on the distinguished side.  We shall refer to this second case 
as the oriented case.  We shall use both versions of the maps. The geometric definition is then as follows. 


Given a smooth hypersurface M ∈^n,  with a smooth normal vector 
field  on M, the (oriented) Lorentz map is the natural map 
: M →^n+1 defined by (x) = z, where to Π = T_xM is 
associated the plane Π^'', Lorentzian orthogonal line 
ℓ, and the distinguished intersection z with ^n+1.  In the general case where we do not have an orientation for M, we define  
: M →^n+1 by (x) is the equivalence 
class of ± z in ^n+1.  

In fact, from the algebraic form of this map to follow, we shall see that it 
actually maps into an n dimensional submanifold ^n of ^n+1.
	We give a specific algebraic form for this map. 


 §.§ Algebraic (Coordinate) Definition of the Lorentzian Map
 
We can give a coordinate definitions for the maps.  If T_xM is defined by 
·  = c, where = (x_1, … , x_n).  Then, 
Π^' contains T_xM and _n+1 and so is defined by 
· + c x_n+1  = c.  Then, Π^'' contains 
Π^'×{_n+2} and the origin so it is defined by 
· + cx_n+1 - c x_n+2 = 0.  Thus, the Lorentzian orthogonal 
line ℓ is spanned by ( , c, c), which we write in abbreviated 
form as ( , c) with = (1, 1).  Hence, the map : M →^n+1 sends x to ( , c), and the general case sends it to 
the equivalence class in ^n determined by ( , c).  We 
shall be concerned with a subspace of ^n+1 where this duality 
corresponds to hypersurfaces of ^n.  The general correspondence is 
used in <cit.> to parametrize (n-1)-dimensional spheres in ^n.  
We need on M a smooth normal unit vector field  orienting M.  
Given the normal vector field , it defines a distinguished side of 
T_xM.
In fact, the image lies in the submanifold ^n of ^n+1 defined 
by 

    ^n    =   { (, c) : ∈ S^n-1 , c ∈}
 
which we can view as a submanifold ^n ⊂^n+1; or in the 
general case it lies in ^n.  


Given a smooth hypersurface M ∈^n,  with a smooth normal vector 
field  on M, the (oriented) Lorentz map is the natural map 
: M →^n defined by (x) = ( , c), where T_xM is 
defined by · = c.  In the general case, we choose a local 
normal vector field  and then (x) is the equivalence class of 
( , c) in ^n.  

In the following we shall generally concentrate on the oriented case and 
the map , with the general case just involving considering the map to 
equivalence classes.  Using  or , we are led to considering the geodesic flow in 
^n+1, and obtain the induced geodesic flow on ^n+1.   
Once we have determined the geodesic flow between points in ^n, 
there 
are two questions concerning  to lift the flow back to hypersurfaces 
in ^n.  One is when  is nonsingular, and at singular points what 
can we say about 
the local properties of  when M is generic.  The second question is 
how we may construct the inverse of  when it is a local embedding 
(or immersion).  



§ LORENTZIAN GEODESIC FLOW ON ^N+1

 
We give the general formula for the geodesic flow between points  z_0 = 
(_0, d_0 ) and z_1 = (_1, d_1 ) in ^n.  

 §.§ Several Auxiliary Functions
 To do so we introduce several auxiliary functions.  
We first define the function (x, θ) by


    (x, θ)      =     {sin(xθ)/sin(θ)     θ≠ 0 
       x      θ = 0
    .

Then, sin(z) is a holomorphic function of z, and the quotient 
sin(xθ)/sin(θ) has removable singularities along 
θ = 0 with value x.  Hence, (z, θ) is a holomorphic 
function of (z, θ) on × ((-π , π) × ), and 
so analytic on × (-π , π).  Also, directly computing the 
derivative we obtain 

    ((x, θ)x     =     {cos( xθ) ·θ/sinθ     θ≠ 0 
       1      θ = 0
    .


In fact, we can recognize (n, θ) for integer values n as the 
characters for the irreducible representations of SU(2)  restricted to 
the 
maximal torus.

We also introduce a second function for later use in <ref>.  
For - π/2 < θ < π/2, we define

    μ (x, θ)    =   cos(xθ)/cos(θ).
 
Then, there is the following relation 

    (x, θ) + (1-x, θ)     =   μ (1 -2x, θ/2)

This follows by using the basic trigonometric formulas  sin (x) + sin 
(y) =  
2 cos (1/2 ( x - y)) sin(1/2 ( x + y)) and sinθ = 
2 
sin (1/2θ)  cos (1/2θ).  There are additional 
relations between these two functions that follow from other basic 
trigonometric identies.




   




 §.§ Geodesic Curves in ^n+1 joining points in ^n
 
We may express the geodesic curve between z_0 = (n_0, c_0 ≥) and 
z_1 = (n_1, c_1 ≥) in 
^n+1 using (x, θ) provided n_1 ≠  -n_0.  
We let - π/2 < θ < π/2 be defined by cosθ 
= _0·_1.  



Provided n_1 ≠  -n_0, the geodesic curve ≫(t) in ^n+1 
between points ≫(0) = z_0 and ≫(1) = z_1 in ^n for the 
Lorentzian metric on ^n+1 is given by

    ≫(t)    =   (t, θ)  z_1   +  (1-t, θ)  z_0   
           0 ≤ t ≤ 1

Furthermore, this curve lies in ^n for 0 ≤ t ≤ 1.  Hence, 
^n is a geodesic submanifold of ^n+1. 

We can expand the expression for ≫(t) and obtain the family of 
hyperplanes Π_t in ^n.   Expanding (<ref>) we obtain

    n_t      =   (t, θ) _1   +  (1-t, θ) _0  
       and 
    
     c_t       =   (t, θ)  c_1   +  (1-t, θ)  c_0

Then the family Π_t is given by 

    Π_t    =   { = (x_1, … , x_n) ∈^n : ·_t 
    = c_t}

We can also  compute the initial velocity for the geodesic in 
(<ref>).  


The initial velocity of the geodesic (<ref>) with θ≠ 0 is 
given by 

    ≫^'(0)    =   θ/sinθ· 
    (__0 (_1), (c_1 - cosθ  c_0)  )
 
where __0 denotes projection along _0 onto the line  
spanned by .  If θ = 0, then _0 = _1 and the velocity 
is 
(0, (c_1 - c_0)) (with Lorentzian speed 0). 


Note that  

    (__0 (_1), (c_1 - cosθ  c_0)  ) _L    = 
      __0 (_1)

which equals sinθ.  We conclude that the Lorentzian magnitude of 
≫^'(0) is θ.  Since geodesics have constant speed, the 
geodesic will travel a distance | θ |.  Hence, | θ | is the 
Lorentzian distance between z_0 and z_1.


Let P be the plane in ^n+1, 1 which contains 0, z_0 and z_1.  
The geodesic curve between z_0 and z_1 is obtained as a constant 
Lorentzian speed parametrization of the curve obtained by intersecting 
P 
with ^n+1.  We choose a unit vector ∈Π such that 
_1 is in 
the plane through the origin spanned by  _0 and .  Let 0 ≤θ < π  be the angle between _0 and _1 so 
cosθ = _0 ·_1.  Then, 
_1 - (_1·_0) _0 is the projection of 
_1 along _0 onto the line spanned by  whose direction is 
chosen so that  _1 - cosθ _0 = sinθ.  Then, a tangent vector to ^n+1∩ P  at the point z_0 is given by 

    (_1 - cosθ _0, (c_1 - cosθ  c_0) )  = (sinθ , (c_1 - cosθ  c_0) )

Then, we seek a Lorentzian geodesic ≫(t) in the plane P beginning at 
(_0, c_0 ) with initial velocity in the direction (sinθ , 
(c_1 - cosθ  c_0) ).  Consider the curve 

    ≫(t)    =    (cos(t θ) _0   +  sin (t θ) , (cos(t 
    θ) c_0   +  sin (t θ)/sin (θ)(c_1 - cosθ  
    c_0)) )

First, note that ≫(0) = z_0, and ≫(1) = z_1.  Also, this curve lies
 in the plane spanned by z_0 and (<ref>).  Also, 

    ≫(t)_L    =   cos(t θ) _0   +  sin (t θ) 
       =    1

as _0 and  are orthogonal unit vectors.  Hence, ≫(t) is a 
curve parametrizing ^n+1∩ P.  It remains to show that 
≫^'' is Lorentzian orthogonal to ^n+1 to establish 
that it is a Lorentzian geodesic from z_0 to z_1.
A computation shows 

    ≫^''(t)    =    - θ^2 ( cos(t θ) _0   + 
     sin (t θ) , sin (t θ)/sin (θ) (c_1 - cosθ  c_0) )

which is - θ^2≫(t), and hence Lorentzian orthogonal to 
^n+1.  
Because of the fraction sin (t θ)/sin (θ), we have to 
note that when θ = 0, then _0 = _1 and ≫(t) takes the 
simplified form

    ≫(t)    =    (_0, c_0   +   t (c_1 - c_0)) )

which is still a Lorentzian geodesic between z_0 to z_1.  Lastly, we must show that this agrees with (<ref>).  First, consider 
the case where θ≠ 0.

    = 1/sinθ  (_1 - cosθ _0)

Substituting this into the first term of the RHS of (<ref>), we 
obtain

    1/sinθ(sinθ cos(t θ) - cosθ sin 
    (t 
    θ)) _0   +  sin (t θ)/sinθ _1

which by the formula for the sine of the difference of two angles  equals 

    sin((1-t)θ)/sinθ_0   +  sin (t 
    θ)/sinθ_1

Analogously, we can compute the second term in the RHS of 
(<ref>),
to be

    sin((1-t)θ)/sinθ  c_0   +  sin (t 
    θ)/sinθ c_1

This gives (<ref>) when θ≠ 0.  When θ = 0, _0 
= _1 and the derivation of (<ref>) from (<ref>) for 
θ = 0 is easier.



We have alread seen that the geodesic flow between the planes 
_0· x = c_0 and _1· x = c_1 induced from the geodesic 
flow in P^n ∨ corresponds to the geodesic flow between (_0, 
c_0) and (_1, c_1), which is given by the unit speed curve in the 
intersection of the plane P, containing these points and the origin, with 
the unit sphere S^n.  If we replaced (<ref>) by linear interpolation

    ≫(t)    =    t  (_1, c_1)   +    (1-t)  (_0, c_0)   
           0 ≤ t ≤ 1

then the curve lies in the plane P and its projection onto the unit sphere 
does parametrize the geodesic, but it is not unit speed, and as we 
remarked earlier it is not invariant under translation and hence not under 
rigid transformations. 



§ INVARIANCE OF LORENTZIAN GEODESIC FLOW AND SPECIAL CASES

 
We investigate the invariance properties of Lorentzian geodesic flows and 
the properties of these flows in special cases.


 §.§ Invariance of Lorentzian Geodesic Flow
 
We first claim the Geodesic flow given in Proposition <ref> is 
invariant under the extended Poincare group generated by rigid 
transformations and scalar multiplications.  By this we mean 
the following.  
If ≫(t) = (_t, c_t) is the Lorentzian geodesic flow between 
hyperplanes P_0 and P_1 defined by _0· x = c_0, 
respectively _1· x = c_1, then ψ̃(≫(t)) is the 
Lorentzian geodesic flow between 
hyperplanes ψ(P_0) and ψ(P_1).


The Lorentzian geodesic flow is invariant under the extended Poincare 
group.


Suppose z_i = (_i, c_i) ∈^n, i = 1, 2, and let Π_i be the 
hyperplane determined by z_i.  Let ψ be an alement of the extended 
Poincare group.  It is a composition of scalar multiplication by b 
followed by a rigid transformation so ψ() = b  A() +, with 
A an orthogonal transformation.  Then, 
Π_i^' = ψ (Π_i) is defined by 

    ψ̃(z_i) = ψ̃(_i, c_i) = (A(_i), bc_i + _i·).















If cos(θ) = _0·_1, then by (<ref>) the 
Lorentzian geodesic flow is given by γ(t) defined by

    (_t, c_t)    =    ((t, θ)_1   +  (1-t, θ) _0,  (t, θ)  c_1   +  (1-t, θ)  c_0 )

defining the family of hyperplanes Π_t.  Then, by (<ref>) 
Π_t^' = ψ(Π_t) is defined by ^'_t· x = 
c^'_t, where ψ̃(γ(t)) is defined by


    (^'_t, c^'_t)      =   ( A((t, θ)_1   
    +  (1-t, θ) _0),   b((t, θ))c_1 + (1-t, θ))c_0)   + . 
                . A((t, θ)_1   +  (1-t, 
    θ)·)) 
       =   (t, θ)(A(_1),   b c_1 + A(_1)·)   
    +  
    (1-t, θ)( A(_0),    b c_0 + A(_0)·) 
    
       =   (1-t, θ)ψ̃(z_1) + (1-t, θ)ψ̃(z_0)


which is the geodesic flow between Π_0^' defined by ψ̃(z_0) and Π_1^' defined by ψ̃(z_1).  



An alternate way to understand Proposition <ref> is to observe 
that the extended Poincare group acts on ^n ×≥ sending 
(, c≥) ↦ (A(), (b c + ·)≥).  This action 
preserves the Lorentzian inner product on this subspace and preserves 
^n.  Hence, it maps geodesics in ^n to geodesics in ^n.  



 §.§ Special Cases of Lorentzian Geodesic Flow

We next determine the form of the Lorentzian geodesic flow in several 
special cases. [Hypersurfaces Obtained by a Translation and Homothety] 
Suppose that we obtain Π_1 from Π_0 by translation by a vector 
 and multiplication by a scalar b.  The correspondence associates 
to ∈Π_0, bx^' =  + ∈Π_1.  Then, the geodesic flow is given by the following.


Suppose Π_0 is the hyperplane defined by _0· x = c_0, with 
_0 a unit vector, and Π_1 is obtained from Π_0 by 
multiplication by the scalar b ≠ 0 and then translation by .  Then 
the Lorentzian geodesic 
flow Π_t is given by the family of parallel hyperplanes defined by 
_0· x = c_t where c_t = (1+ (b - 1)t)c_0 + t _0·.  


If  Π_0 is defined by _0· x = c_0, with _0 a unit 
vector, then, Π_1 is defined by _1· x = c_1 where _1 = 
_0 and c_1 = b c_0 + _0 ·.  Thus, Π_1 is parallel to 
Π_0.  Thus, as _1 = _0, θ = 0 and (t, 0) = t, so the geodesic 
flow Π_t is given by 

    t (_0, c_1 )  + (1-t) (_0, c_0 )       =     (_0, ((1-
    t)c_0 + t (b c_0 + _0 ·))  ) 
         =    (_0, ((t b + 1-t)c_0 + t (_0 ·)) )

so that Π_t is defined by _0· = c_t where c_t = (1 + (b 
- 1)t)c_0 + t (_0 ·).  

This defines a family of hyperplanes parallel to Π_0 where 
derivative of the translation map is the identity; hence, under translation 
_0 is mapped to itself translated to ^' =  +.  
Thus, under the correspondence, _1 = _0.  Also, If _0 · = c_0 is the equation of the tangent plane for M_0 at a point , 
then the tangent plane of M_1 at the point ^' is 

    _1 ·^'  =  _0 ·  ( + )   =   c_0 + 
    _0 ·

Hence, c_1 = c_0 + _0 ·.  As _0 = _1, θ = 0.  Thus the geodesic flow on ^n is 
given 
by 

    t (_0, c_1 )  + (1-t) (_0, c_0 )     =     (_0, c_0 
    ) + (0, (t _0 ·)  )     =    (_0, (_0 · ( 
    +  t )) )

Thus, at time t the tangent space is translated by t.  Thus, the 
envelope of these translated hyperplanes is the translation of M_0 by t.  


If a hypersurface M_1 is obtained from the hypersurface M_0 by a 
translation combined with a homothety ^' = ψ() = b  +, then for each x ∈ M_0 with image x^'∈ M_1 the 
Lorentzian geodesic flow will send the tangent plane T_xM_0 to the 
tangent plane T_x^'M_1  by the family of parallel hyperplanes 
given by Corollary <ref>.  Thus, for each 0 ≤ t ≤ 1, the 
hyperplane under the geodesic flow will be the tangent plane 
T__tM_t, where for ψ_t() = (1 + (b - 1) t)  + t, 
M_t = ψ_t(M_0) and _t = ψ_t() ∈ M_t.  Thus, the Lorentzian 
geodesic flow will send M_0 to the family of hypersurfaces 
M_t = ψ_t(M_0). 



[Hyperplanes Obtained by a Pseudo-Rotation]
 
Second, suppose that Π_0 and Π_1 are nonparallel affine hyperplanes.  
Then, W = Π_0 ∩Π_1 is a codimension 2 affine subspace.  The 
unit normal vectors _0 and _1 lie in the orthogonal plane 
W^⊥ with _0 ·_1 = cos(θ) with -π/2 < 
θ < π/2.  Since the Lorentzian geodesic flow commutes with 
translation, we may translate the planes and assume that W contains 
the origin.  Then, both c_0 and c_1 equal 0.  Thus, by Proposition 
<ref>, the Lorentzian geodesic flow from Π_0 to Π_1 is 
given by (_t, c_t) for 0 ≤ t ≤ 1, where

    (_t, c_t)     =     ((t, θ)  (_1, c_1)   +  (1-t, 
    θ)  (_0, c_0)

Thus, _t = (t, θ) _1 + (1-t, θ) _0, while c_t ≡ 
0.  Hence, the hyperplane Π_t is defined by _t · = 0 so it 
contains W.  However, its intersection with the plane W^⊥ is the 
line orthogonal to _t, which by the above expression for _t, does 
not give a standard constant speed rotation in the plane.  We refer to this 
as a pseudo-rotation. 
Instead consider a rotation A of hyperplanes Π_0 to Π_1 about 
an axis not containing W.  We consider the form of the pseudo-rotation.  
As an example, consider the case of a rotation A about the origin in a 
plane (which pointwise fixes an orthogonal subspace.  Choosing 
coordinates, we may assume that the rotation A is in the 
(x_1, x_2)–plane and rotates by an angle .  We suppose Π_0, 
defined by _0 · = c_0.  As A (_0) · A() = _0 
· = c_0, if we let ^' = A(), then the equation of 
the hyperplane Π_1  is defined by A (_0) ·^' = 
c_0.  
Hence, _1 = A (_0) and c_1 = c_0.  To express the geodesic flow, we write _0 =  + where  
is in the rotation plane and  is fixed by A.  Hence, _1 = A() 
+.  Thus, the angle θ between _0 and _1 satisfies 

    cosθ  =  _1 ·_0   =   A()·  +  ·

As _0  = 1, we obtain ·  +  · = 1.  
Also, A()· = ^2 cos.   Hence, 

    cosθ  =    1 + ^2 (cos - 1)

We recall that by (<ref>)


    (t, θ) + (1-t, θ)     =   μ (1 -2t, θ/2)


Using the expressions for _0 and _1, we find the geodesic flow 
is 
given by

    =   (t, θ)  (A(_0), c_0 )  +  (1-t, θ)  (_0, 
    c_0 )  
       =     
    (((t, θ) A() + (1-t, θ) )   +  μ (1 -2t, 
    θ/2) ,   μ (1 -2t, θ/2) c_0 )

We note that μ (1 -2t, θ/2) is a function of t on 
[0, 1] which has value 
= 1 at the end points, and has a maximum =  (1/2θ) 
at t = 1/2.  Thus, the geodesic flow (_t, c_t) has the 
contribution in the rotation plane given by 
(t, θ) A() + (1-t, θ) which is not a true rotation 
from  to A().  Also, the other contribution to _t is from 
μ (1 -2t, θ/2) which increases and then returns to 
size  (see Fig. <ref>).  In addition, the distance from the origin 
will vary by μ (1 -2t, θ/2) c_0.  This is the form of the  
pseudo rotation from Π_0 to Π_1.  
This yields the following corollary.


If Π_1 is obtained from Π_0 by rotation in a plane (with fixed 
orthogonal complement), then the Lorentzian geodesic flow is the family 
of hypersurfaces obtained by applying to Π_0 the family of pseudo 
rotations given by (<ref>).  





§ FAMILIES OF LORENTZIAN PARALLEL FRAMES ON LORENTZIAN GEODESIC 
FLOWS


A Lorentzian geodesic flow from hyperplanes Π_0 to Π_1 may be 
viewed as a minimum twisting family of hyperplanes Π_t joining 
Π_0 to Π_1.  If in addition, we are given orthonormal frames 
{ e_i  0} for Π_0 and { e_i  1} for Π_1, we ask what 
form a minimum twisting family of smoothly varying frames { e_i  
t} for Π_t should take?  We give the form of the family of Lorentzian parallel  orthonormal frames in Π_t beginning with 
{ e_i  0}, and then use this family to construct a family of frames 
from { e_i  0} to { e_i  1} which can be made to satisfy 
various criteria for minimal Lorentzian twisting.  





 §.§ Criterion for Lorentzian Parallel Vector Fields
  

Given a smooth curve ≫(t), 0 ≤ t ≤ 1 in ^n and a smoothly 
varying  family of affine hyperplanes {Π_t} satisfying: 


  1) ≫(t) ∈Π_t for each t;

  2) ≫(t) is tranverse Π_t for each t.

We let _t denote the smooth family of unit normals to the hyperplanes 
Π_t.  Then there is a corresponding curve in ^n+1 defined by 
≫̃(t) = (_t, c_t ) where c_t = <≫(t), _t>.  Let 
_t denote a smooth section of {Π_t}, by which we mean that if 
we view _t as a vector from the point ≫(t) lies in the hyperplane 
Π_t for each t.  There is then a corresponding vector field 
_t on ≫̃(t)defined by _t = (_t, (t) ).  
This vector field is tangent to ^n+1 as the vector N_t = (_t, c_t ) 
is Lorentzian normal to ^n+1 at ≫̃(t) so < N_t, _t>_L = <_t, _t> = 0.  We give a criterion for _t to be a Lorentzian parallel vector field along 
≫̃(t). 
[Criterion for Lorentzian Parallel Vector Fields]

The smooth vector field _t is Lorentzian parallel along ≫̃(t) if :


  i)  e_tt = φ(t) _t for a smooth function φ(t)

  ii) (t) = ∫φ(t) c_t  dt for each t.



As N_t is Lorentzian normal to ^n+1 at ≫̃(t), it is sufficient to show that ẽ_tt = (t) N_t for some function (t).  Then, by i) and ii)

    ẽ_tt     =    (e_tt, ^'(t))    =    (φ(t) _t, ^'(t) )  
         =   φ(t) (_t, c_t )    =   φ(t) N_t

Hence, ẽ_t is Lorentzian parallel.

Hence, a smooth section _t of {Π_t} extends to a Lorentzian parallel vector field _t provided condition i) is satisfied and using condition ii) to define 
(t). 



Suppose Π_t is the normal hyperplane to ≫(t) at the point ≫(t) for each t.  Then the condition that _t is a section of Π_t is that <_t, ≫^'(t)> = 0.  Then, by Lemma <ref> the condition that _t is moreover a parallel vector field is that there is a smooth function φ(t) so that _tt = φ(t) ≫^'(t).  These two conditions are the criteria in <cit.> and other papers quoted there that for the normal family of affine planes in ^3 the vector field _t has minimum rotation.   


In the case when the family of affine hyperplanes {Π_t} are not normal, then the vectors _t and ≫^'(t) are not parallel so the condition in Lemma <ref> replaces the role  of ≫^'(t) in both conditions by _t.


Then, for each such vector field ζ(t) and smooth function (t), 
we define a smooth tangent vector field to ^n+1 (in fact ^n) 
along ≫(t) by ζ̃(t) = (ζ(t), (t) ).  We 
observe that at each point (_t, c_t ), ζ̃(t) is 
Lorentzian orthogonal to (_t, c_t ) and so is tangent to 
^n+1.  Moreover, because of the form of ζ̃(t), it is also 
tangent to ^n.  However, there may be no specific choice of (t) 
possible for ζ̃(t) to be a Lorentzian parallel vector field on 
^n.  We say that the smooth section ζ(t) of {Π_t} is a 
Lorentzian parallel vector field if there is a smooth function 
(t) so that ζ̃(t) is a Lorentzian parallel vector field on 
^n.  For example, in the special case that the section ζ(t) ≡ is 
constant we may choose (t) ≡ 0, and the vector field ζ̃(t) = (ζ(t), 0·) is constant and so is a Lorentzian 
parallel vector field.  
Thus, given a set of such vector fields {ζ_i(t): i = 1, … , k}  
which are sections of {Π_t} together with smooth functions 
_i(t), then we obtain a set of vector fields {ζ̃_i(t): i = 1, 
… , k} on ≫(t) tangent to ^n.  Then, the existence of 
Lorentzian parallel families of frames for 
{Π_t}is given by the following.


Let ≫(t) = (_t, c_t ) be a Lorentzian geodesic defining the 
family of hyperplanes {Π_t} in ^n.  If { e_i  0, 1 ≤ i ≤ 
n-1} is an orthonormal frame for Π_0, there is a (smoothly varying) 
family of orthonormal frames { e_i  t, , 1 ≤ i ≤ n-1} for {Π_t} such that the vector fields {ẽ_i  t, 1 ≤ i ≤ n-
1} form a family of Lorentzian parallel vector fields on ^n which 
are Lorentzian orthonormal.  


First, if Π_0 and Π_1 are parallel then Π_1 is a translation of 
Π_0, so by Corollary <ref> the Lorentzian geodesic flow 
Π_t is a family of hyperplanes parallel to Π_0 so the family of 
frames is the constant  family obtained by translating { 
e_i  0} to each hyperplane in the family.  The corresponding family 
{ẽ_i  t} is also constant, and hence Lorentzian parallel. 
Next we consider the case where Π_0 and Π_1 are not parallel.  We 
first construct the required Lorentzian parallel family beginning with a 
specific orthonormal frame for Π_0.  Then, we explain how to modify 
this for a general orthonormal frame.  We have Π_0 is defined by _0· = c_0, and Π_1, by 
_1· = c_1 with _0·_1 = cos(θ) for -
π/2 < θ < π/2.  Then, as earlier in <ref>, W = 
Π_0 ∩Π_1 is a codimension 2 affine subspace, and every 
hyperplane Π_t in the Lorentzian geodesic flow ≫(t) = (_t, c_t 
) from Π_0 to Π_1 contains W.  We let e_2, … , e_n-1 denote an orthonormal frame for W.  Then, 
the e_i define constant vector fields e_i along the Lorentzian 
geodesic with each e_i ∈ W ⊂Π_t. These allow us to define 
ẽ_i, 





which are parallel vector fields on ^n+1 (in fact ^n along the 
Lorentzian geodesic ≫(t).  Hence, to complete them to an orthonormal frame, we need only construct a 
unit vector field e_1, t which is a smooth section of {Π_t} 
orthogonal to W for each 0 ≤ t ≤ 1 and show that the resulting 
vector field ẽ_1, t is a Lorentzian parallel vector field on 
≫(t).  The subspace of any Π_t orthogonal to W is one dimensional, so 
there are two choices for a unit vector spanning it.   For Π_0 we 
choose e_1, 0 so that _0, e_1, 0, e_2, … , e_n-1 is 
positively oriented for ^n.  Likewise we choose e_1, 1 for Π_1 
so that _1, e_1, 1, e_2, … , e_n-1 is positively oriented for 
^n.  Then, we define 

    e_1, t   =   (t, θ) e_1, 1  +  (1-t, 
    θ) e_1, 0

We first claim that e_1, t is a unit vector field such that e_1, t∈Π̃_t for all 0 ≤ t ≤ 1.  That e_1, t is a unit vector field follows from the calculation for 
_t in Proposition <ref>.  Second, we compute 

    e_1, t·_t      =    ((t, θ) e_1, 1  +  (1-t, 
    θ) e_1, 0) · ((t, θ) _1   +  (1-t, θ) _0)  
    
       =    ((t, θ) (1-t, θ)) (e_1, 1·_0 + e_1, 0·_1)

To see the RHS of (<ref>) is zero, we consider the two positively 
oriented orthonormal bases for W^⊥: _0, e_1, 0 and _1, 
e_1, 1.  If we represent _1 = a _0 + b e_1, 0, then necessarily 
e_1, 1 = -b _0 + a e_1, 0.  Then, 

    e_1, 1·_0 + e_1, 0·_1    =    -b + b     =    
    0

We also note that e_1, t is orthogonal to W̃ for all t.  Thus, 
the resulting tangential vector fields ẽ_1, t, ẽ_2, … , 
ẽ_n-1 are mutually Lorentz orthogonal and are Lorentzian unit 
vector fields.  The vector fields ẽ_i, i = 2, … , n-1 are 
constant and hence Lorentzian parallel.  It remains to show that ẽ_1, t is Lorentzian parallel.  We claim that if (t) = 
1/θ c^'_t, then ẽ_1, t = (e_1, t, (t) 
·≥) is a Lorentzian parallel tangent vector field along ≫(t).  
As ≫(t) is a Lorentzian geodesic, ≫^'(t) is Lorentzian 
parallel along ≫(t).  We will show that with the given (t), ẽ_1, t = 1/θ≫^'(t).  




From the proof of Proposition <ref>, by (<ref>), ≫(t) 
can be written 

    ≫(t)    =    (cos(t θ) _0   +  sin (t θ) , (cos(t 
    θ) c_0   +  sin (t θ)/sin (θ)(c_1 - cosθ  
    c_0)) )

Hence, 

    ≫^'(t)    =   θ (-sin(t θ) _0   +  cos (t 
    θ) , (-sin(t θ) c_0   +  cos (t θ)/sin (θ)(c_1 - cosθ  
    c_0)) )

Both {_0, ^⊥_0} and {_1, ^⊥_1} have 
positive orientation in W^⊥; hence e_1, 0 = ^⊥_0 and 
e_1, 1 = ^⊥_1   If we represent _1 = a _0 + b 
^⊥_0, then ^⊥_1 = -b _0 + a ^⊥_0 and in 
(<ref>) the unit vector = ^⊥_0 if b > 0 and -
^⊥_0 if b < 0.  Second we represent e_1, t in the same form as (<ref>).  To do 
so we compute the unit vector in the same direction as the projection of 
e_1, 1 (= ^⊥_1) along e_1, 0 (= ^⊥_0).  
Then, by the above, e_1, 1 = -b _0 + a e_1, 0.  Thus, the 
corresponding  for this case is either _1 =  - ^⊥_0 if 
b > 0 or ^⊥_0 if b < 0.  Thus, by the calculation in the proof 
of Proposition <ref>, in either case we obtain

    e_1, t     =   cos (t θ) ^⊥_0    +  sin(t θ) 
    e_1, 0
         =    -sin(t θ) _0   +  cos (t θ) ^⊥_0

This equals the first component of (<ref>), which implies the 
equality e_1, t = 1/θ^'_t.  The last step is to obtain the result for any orthonormal frame {f_i, 
0} in Π_0.  

There is an orthogonal transformation A so that A(e_i, 0) = f_i, 0.  
If we express f_i, 0 = ∑_j = 1^n-1 a_i, j e_j, 0.  Then, we can 
define vector fields f̃_i, t = ∑_j = 1^n-1 a_i, jẽ_j, t.  Since the f̃_i, t are constant linear combinations of 
Lorentzian parallel vector fields,and hence are Lorentzian parallel 
themselves.  As they are obtained by an orthogonal transformation of an 
orthonormal frame field, they also form an orthonormal frame field. 



 §.§ Interpolating between Orthonormal Frames
 
Now we consider given frames {e_i, 0} in Π_0 and {f_i, 1} 
in Π_1, such that {_0, e_1, 0, … , e_n-1, 0} and 
{_1, f_1, 0, … , f_n-1, 0} have the same orientation (which 
we may assume are positive.  We may first construct the Lorentzian 
parallel family of orthonormal frames {e_i, t}.  Then, the smoothly 
varying family {_t, e_1, t, … , e_n-1, t} will retain positive 
orientation for each t.  Hence, {e_i, 1} and {f_i, 1} have the 
same orientation.  Thus, there is an orthogonal transformation B of 
Π_1 such that B(e_i, 1) = f_i, 1 and (B) = 1.  Again we may 
represent B using the basis {e_i, 1} by an orthogonal matrix b_i, 
j.  As (B) = 1, there is a one parameter family exp(s E) so that 
exp( E) = B for a skew symmetric matrix E.  Then, given a smooth 
nondecreasing function φ : [0, 1] → [0, 1] with φ(0) = 0 
and φ(1) = 1, we can modify the Lorentzian parallel family {e_i, 
t} by 
{exp(φ(t) E)(e_i, t)}, which is a family of orthonormal 
frames.  In this family we see that the total amount of 
twisting  from Lorentzian parallel family is given by the 
orthogonal transformation B (or skew-symmetric matrix E).  The 
introduction of the twisting in the family is given by the function 
φ.   
[Planes in ^3]

In the case of planes Π_0 and Π_1 in ^3 with _0 ≠±_1 and _0 ·_1 = cos(θ), we can easily construct 
the family of Lorentzian parallel frames by letting e_2, t be a 
constant unit vector field in the direction of _0 ×_1, and 
ẽ_1, t = 1/θ^'_t for the Lorentzian 
geodesic flow ≫(t) = (_t, c_t) from Π_0 to Π_1.  Then, 
{e_1, t, e_2, t} gives a Lorentzian parallel family of frames.  If e^'_1, e^'_2 is another frame for Π_0 with the 
same orientation as e_1, 0, e_2, 0, then there is a rotation with 
rotation matrix R so that Re_i, 0 = e^'_i.  Then {R e_1, t, 
R e_2, t} gives a Lorentzian parallel family of frames beginning 
with e^'_1, e^'_2.   Furthermore, if { f_1, f_2} is a 
positively oriented frame for Π_1, then there is a rotation matrix 
S_ by an angle  so that S_R e_i, 1 = f_i.  Then, for (t), 
0 ≤ t ≤ 1, a nondecreasing smooth function from 0 to 1, the 
family of rotations S_(t) provides an interpolating family 
{S_(t)R e_1, t, S_(t)R e_2, t} from {e_1, 0,  e_2, 
t} to { f_1, f_2}.  The flexibility in the choice of (t) allows 
for many criterion to be included in choosing the interpolation. [Interpolation for Modeling Generalized Tube Structures]

Generalized tube structures for a region  can be modeled as a disjoint 
union = ∪_t _t of planar regions _t ⊂Π_t for a family 
of hyperplanes {Π_t} along a central curve ≫(t).  The geometric properties 
and structure of the tube can be computed using a smoothly varying family of 
frames e_j, t for {Π_t} (see e.g. <cit.> and <cit.>).  
	This is used in <cit.> for the 3-dimensional modeling of the human 
colon, where normal planes to an identified central curve are modified in high 
curvature regions to form a Lorentzian geodesic, and the family of frames with 
minimal twisting in the sense of Example <ref> are extended to a 
Lorentzian parallel family of frames in the modified family of planes.  This structure 
can then be deformed in various ways for better visualization.


[Family of Normal Planes to a Curve in ^3]

A second situation is for a regular unit speed curve (t) in ^3 
with ≠ 0.  Then there is the Frenet frame {, , } 
defined along (t).  Then, {, } provides a family of 
orthonormal frame for the planes Π_t passing through and orthogonal 
to (t).  The Lorentzian map for the family of planes is given by 
≫(t) = ((t), ·^').  Then,  

    ≫^''(t) = ^' + (- T + τ B), 
    d^2 /d^2t(·^') )

Thus, for this family to be a Lorentzian geodesic family of planes, 
≫^''(t) must be Lorentzian orthogonal to ^4.  For 
this, we must have that the first term is a multiple of , which 
implies ^', τ≡ 0.  Thus,  is a plane curve with 
constant curvature , so it is a portion of a circle and ·^'≡ 0.  Then, B is constant so it is Lorentzian parallel, 
and ≫^''(t) = (-^2 T, 0 ) = - (- T, 0 ), and 
so it follows that (, 0) is Lorentzian parallel.  Hence, {, 
} is a Lorentzian parallel family of orthonormal frames.  We 
summarize this with the following


If (t) is a regular unit speed curve in ^3 with ≠ 0, then 
the family of normal planes Π_t to (t) is a Lorentzian geodesic 
family iff (t) is a portion of a circle.  In this case the Frenet vector 
fields {, } forms a Lorentzian parallel family of orthonormal 
frames in Π_t. 






§ DUAL VARIETIES AND SINGULAR LORENTZIAN MANIFOLDS

 
Before continuing with the analysis of the geodesic flow in  ^n+1 
and the induced flow between hypersurfaces in ^n, we first explain 
the relation of the Lorentzian map with a corresponding map to the dual 
projective space.


 §.§ Relation with the Dual Variety
 Suppose that M ⊂^n is a smooth hypersurface.  There is a 
natural way to associate a corresponding dual varietyM^∨ in the dual projective space P^n ∨ (which consists 
of lines through the origin in the dual space ^n+1  *).  Given a 
hyperplane Π⊂^n,  it is defined by an equation ∑_i 
=1^n a_i x_i = b.  We associate the linear form : ^n+1→ 
defined by (x_1, … , x_n+1) = ∑_i =1^n a_i x_i - bx_n+1.  
As the equation for Π is only well defined up to multiplication by a 
constant, so is , which defines a unique line in ^n+1  *.  This 
then defines a dual mapping : M → P^n ∨, sending x 
∈ M to the dual of T_xM.  In the context of algebraic geometry in the complex case, this map actually 
extends to a dual map for a smooth codimension 1 algebraic subvariety 
M ⊂ P^n, and then the image M^∨ =  (M) is again a 
codimension 1 algebraic subvariety of P^n ∨.  There is an 
inverse dual map ^∨ for smooth codimension 1 algebraic 
subvarieties of P^n ∨ to P^n defined again using the 
tangent spaces.  Hence, ^∨ : M^∨→ P^n.  It is only 
defined on smooth points of M^∨ (which may have singularities); 
however it extends to the singular points of M^∨ and its image is 
the original M.  In our situation, we are working over the reals and moreover M will not 
be defined algebraically.  Hence, we need to determine what properties both 
 and M^∨ have.  We also will explain the relation with the 
Lorentz map.  


 §.§ Legendrian Projections
 
Given M, we let P(^n+1 *) denote the projective bundle over ^n 
given by ^n× P^n ∨, where as earlier P^n ∨ denotes 
the dual projective space.  Then, we have an embedding i : M → 
P(^n+1 *), where i(x) = (x, <_x>), with _x the linear form 
associated to T_xM as above.  We let M̃ = i(M).  There is a 
projection map π : P(^n+1 *) → P^n ∨.  Then, by results 
of Arnold <cit.>, π is a Legendrian projection, and for generic M, 
M̃ is a generic Legendrian submanifold of P(^n+1 *) and the 
restriction π | M̃ : M̃→ P^n ∨ is a generic 
Legendrian projection.  This composition π | M̃∘ i is exactly 
.  Hence, the properties of  are exactly those of the Legendrian 
projection.  In particular, the singularities of M^∨ = π (M̃) are 
generic Legendrian singularities, which are the singularities appearing in 
discriminants of stable mappings, see <cit.> or <cit.>.  
In the case of surfaces in ^3, these are: cuspidal edge, a swallowtail, 
transverse intersections of two or three smooth surfaces, and the 
transverse intersection of a smooth surface with a cuspidal edge  (as 
shown in Fig. <ref>).  The characterization of these singularities implies 
that as we approach a singular point from one of the connected components, 
then there is a unique limiting tangent plane, and in the case of the cuspidal 
edge or swallowtail, the limiting tangent plane is the same for each component.  
Hence, for generic smooth hypersurfaces M ⊂^n, the inverse 
dual map ^∨ extends to all of M^∨, and again will have 
image M.  



 
 











Finally, we remark about the relation between the dual variety M^∨ 
and the image M_ =  (M) (or M_ = (M)).  
To do so, we introduce a mapping involving P^n ∨ and ^n.    
In P^n ∨, there is the distinguished point ∞ = <(0, …, 0, 1)>.  
On P^n ∨\{∞}, we may take a point 
<(y_1, … , y_n, y_n+1)>, and normalize it by 

    (y_1^', … , y_n^', y_n+1^') = c· (y_1, 
    … , y_n, y_n+1),     c = (∑_i = 1^n 
    y_i^2)^-1/2.
  
Then, _y = (y_1^', … , y_n^') is a unit vector.  We 
then define a map ν :  P^n ∨\{∞}→^n sending <(y_1, … , y_n, y_n+1)> to (_y, y_n+1^').  This is only well-defined up to multiplication by -1, which is why we 
must take the equivalence class in the pair of points. If we are on a region 
of P^n ∨\{∞} where we can smoothly choose a 
direction for each line corresponding to a point in P^n ∨, then as 
for the case of the Lorentzian mapping, we can give a well-defined map to 
^n.  This will be so when we consider M^∨ for the oriented case.  
In such a situation, when the smooth hypersurface M has a smooth unit 
normal vector field , it provides a positive direction in the line of 
linear forms vanishing on T_xM.  Then, we have the following relations.  


The smooth mapping ν̃:  P^n ∨\{∞}→^n is a diffeomorphism.

Second, there is the relation between the duality map   and the 
Lorentz map  (or ).  


If M ⊂^n is a smooth hypersurface, then the diagram 
(<ref>) commutes, i.e. 
ν̃∘ =.  If, in addition, M has a smooth unit 
normal vector field , then there is the oriented version of diagram 
(<ref>),  ν∘ =.  

    M    ⟶    P^n ∨
         ↘   ↓ν̃
          ^n



















As a consequence of these Lemmas and our earlier discussion about the 
singularities of M^∨, we conclude that M_ (or 
M_) have the same singularities.  Thus, we may suppose they are 
generic Legendrian singularities.  

Although by Lemma <ref> P^n ∨\{∞} is diffeomorphic to ^n, the first space has a natural 
Riemannian structure while on ^n we have a Lorentzian metric.  
Hence, ν̃ is not an isometry and does not map geodesics
 to geodesics.



There is a natural inverse to ν̃ defined as follows:  
If z = (, c) and = (a_1, … , a_n), then we map z to 
<(a_1, … , a_n, -c)>.  We note that replacing z by -z does not 
change the line <(a_1, … , a_n, -c)>.  This gives a well-defined 
smooth 
map ^n → P^n ∨\{∞} which is 
easily 
checked to be the inverse of ν̃.
 

If T_xM is defined by · = c with = (a_1, … , a_n), 
then (x) = < (a_1, … , a_n, -c) >.  Then, as = 1, ν̃( < (a_1, … , a_n, -c) >) = (a_1, … , a_n, c, c) = (,  c ), 
which is exactly (x).
 


 §.§ Inverses of the Dual Variety and Lorentzian Mappings
 
We consider how to invert both  and .  We earlier 
remarked 
that in the complex algebraic setting, the inverse to  is again a dual 
map ^∨.  As ν̃ is a diffeomorphism, and diagram 
<ref> commutes, inverting  is equivalent to inverting .  Also, constructing an inverse is a local problem, so we may as well 
consider the oriented case.  


Let M ⊂^n be a generic smooth hypersurface with a smooth unit 
normal vector field .  Suppose that the image M_ under  
is a smooth submanifold of ^n.  Then, M is obtained as the envelope 
of the collection of hyperplanes defined by · = c for 
(x) = (, c).


We consider an (n-1)-dimensional submanifold of ^n parametrized 
by u ∈ U given by ( (u), c(u)).  The collection of hyperplanes are 
given by Π_u defined by F(, u) =  (u)· - c(u) = 0. 
Then, the envelope is defined by the collection of equations 
F_u_i = 0, i = 1, … , n-1 and F = 0.  This is the system of linear equations 

    i)    (u)· = c(u)       ii)   _u_i (u)· = c_u_i(u),    i = 1, … , n-1


A sufficient condition that there exist for a given u a unique solution to 
the system of linear equations in  is that the vectors 
, _u_1, … , _u_n-1 are linearly independent.  Since 
_u_i = - S(u_i), for S the shape operator for M, linear 
independence is equivalent to S not having any 0-eigenvalues.  
Thus,  is not a parabolic point of M.  For generic M, the set of 
parabolic points is a stratified set of codimension 1 in M.  Thus, off 
the image of this set, there is a  unique point in the envelope.  Also, if we differentiate equation (<ref>)-i) with respect to u_i 
we 
obtain 

    _u_i (u)·  +   (u)·_u_i    =    c_u_i(u)

Combining this with (<ref>)-ii), we obtain 

    (u)·_u_i    =    0,

and conversely, (<ref>) for i = 1, … , n-1 and (<ref>) 
imply (<ref>)-ii).  Thus, if we choose a local parametrization of 
M 
given by (u), then as (u) is a point in its tangent space, it 
satisfies (<ref>)-i), and hence (<ref>), and also  being a 
normal vector field  implies that (<ref>) is satisfied for all i.  
Thus, (<ref>)-ii) is satisfied.  Hence, M is part of the envelope.  
Also, for generic points of M, by the implicit function theorem, the set 
of 
solutions of (<ref>) is locally a submanifold of dimension n-1.  
Hence, in a neighborhood of these generic points of M, the envelope is 
exactly M.  Hence, the closure of this set is all of M and still consists 
of 
solutions of (<ref>).  Thus, we recover M.  Second, to see that the equations (<ref>) describe the inverse of 
the 
dual mapping, we note by Lemmas <ref> and <ref> that 
ν̃ is a diffeomorphism, ^-1 = ^-1∘ν̃, and the preceding argument gives the local inverse to .
 




§ SUFFICIENT CONDITION FOR SMOOTHNESS OF ENVELOPES

 


To describe the induced geodesic flowbetween hypersurfaces 
M_0 and M_1 in ^n, we will use the Lorentzian geodesic flow in 
^n and then find the corresponding flow by applying an inverse to 
.  
We begin by constructing the inverse for a (n-1)-dimensional manifold 
in ^n parametrized by ( (u), c(u) ), where 
u = (u_1, … , u_n-1).  We determine when the associated family of 
hyperplanes Π_u = {∈^n: (u) · = c(u)} has as an 
envelope a smooth hypersurface in ^n.  We introduce a family of vectors in ^n+1 given by (u) = 
((u), -c(u)).  We also denote u_i by _u_i.  
Next we consider the n-fold cross product in ^n+1, denoted by 
v_1 × v_2 ×⋯× v_n, which is the vector in 
^n+1 whose i-th coordinate is (-1)^i+1 times the n × n 
determinant obtained from the entries of v_1, …, v_n by removing 
the 
i-th entries of each v_j.  Then, for any other vector v, 

    v · (v_1 × v_2 ×⋯× v_n)    =   (v, 
    v_1, …, v_n)

We let 

    =   ×_u_1×⋯×_u_n-1

We let H() denote the (n-1) × (n-1) matrix of vectors 
_u_i   u_j. Then we can form H()· to 
be the (n-1) × (n-1) matrix with entries _u_i   
u_j·.  
Then, there is the following determination of the properties of the 
envelope 
of {Π_u}.


Suppose we have an (n-1)-dimensional manifold in ^n parametrized 
by ( (u), c(u) ), where u = (u_1, … , u_n-1).  We let 
{Π_u} denote the associated family of hyperplanes.  Then, the 
envelope of {Π_u} has the following properties.


  i)  There is a unique point _0 on the envelope corresponding to 
u_0 provided (u_0), _u_1(u_0), …, _u_n-1(u_0) are 
linearly independent.  Then, the point is the solution of the system of equations (<ref>).

  ii)  Provided i) holds, the envelope is smooth at _0 provided 
H()· is nonsingular for u = u_0.

  iii)  Provided ii) holds, the normal to the surface at _0 is 
(u_0) and Π_u_0 is the tangent plane at _0.







We use the line of reasoning for Proposition <ref>.  the condition 
that a point _0 belong to the envelope of {Π_u} is that it satisfy 
the system of equations (<ref>).  A sufficient condition that these 
equations have a unique solution for u = u_0 is exactly that (u_0), 
_u_1(u_0), …, _u_n-1(u_0) are linearly independent.   Furthermore, if this is true at u_0 then it is true in a neighborhood of 
u_0.  Thus, we have a unique smooth mapping (u) from a 
neighborhood of u_0 to ^n.  By the argument used to deduce 
(<ref>), we also conclude

    (u)·_u_i    =    0,       i = 1, …, n-1

Hence, if (u) is nonsingular at u_0, then (u_0) is the normal 
vector to the envelope hypersurface at _0, so the tangent plane is 
Π_u_0.  Thus iii) is true. It remains to establish the criterion for smoothness in ii).  As earlier 
mentioned the envelope in the neighborhood of a point _0 is the 
discriminant of the projection of V = {(, u) : F(, u) =  (u)· - c(u) = 0} to ^n.  It is a standard classical result that at a point 
(_0, u_0) ∈ V, which projects to an envelope point _0, the 
envelope is smooth at _0 provided (_0, u_0) is a regular point 
of F (so V is smooth in a neighborhood of (_0, u_0)) and the partial 
Hessian (^2Fu_i  u_j(_0, u_0)) is nonsingular.  For our 
particular F this Hessian becomes H()·_0 - H(c), where 
H() is the (n-1) × (n-1) matrix of vectors (_u_i   u_j), and H()·_0 denotes the (n-1) × (n-1) matrix whose entries 
are _u_i   u_j·_0.  This can also be written H()·_0, where _0 is the extension of _0 to ^n+1 by adding 
1 as the n+1-st coordinate. Now _0 is the unique solution of the system of linear equations 
(<ref>).  This solution is given by Cramers rule.   Let N(u_0) 
denote the n × n matrix with columns (u_0), _u_1(u_0), 
… , _u_n-1(u_0).  By Cramers rule, if we multiply 
_0 by (N(u_0)) we obtain (-1)^n.  Thus, 
multiplying H()·_0 by (N(u_0)) yields 
(-1)^n H()·.  Hence, the nonsingularity of 
H()· implies that of (^2Fu_i  u_j(_0, u_0)).


Although Proposition <ref> handles the case of a smooth manifold 
in ^n, we saw in <ref> that usually the image in 
^n of a generic 
hypersurface M in ^n will have Legendrian singularities and the 
image itself is a Whitney stratified set M̃.  Next, we deduce the 
condition ensuring that the  envelope is smooth at a singular point 
_0.  Because M̃ has Legendrian singularities, it has a special property.  
To expain it we use a special property which holds for certain Whitney 
stratified sets.

  An m-dimensional Whitney stratified set M ⊂^k 
has the Unique Limiting Tangent Space Property (ULT property) if 
for any x ∈ M_sing, a singular point of M, there is a unique 
m-plane Π⊂^k such that for any sequence { x_i } of 
smooth points in M_reg such that lim x_i = x, we have 
lim T_x_iM = Π



For a generic Legendrian hypersurfaces M ⊂^n, if z ∈M̃, then 
M̃ can be locally represented in a neighborhood of z as a finite 
transverse union of (n-1)-dimensional Whitney stratified sets Y_i 
each 
having the ULT property.

Transverse union means that if W_i j is the stratum of Y_i 
containing 
z than the W_i j intersect transversally.   The Lemma follows because M̃ consists of generic 
Legendrian singularities, which are either stable (or topologically stable) 
Legendrian singularities.  These are either discriminants of stable 
unfoldings of multigerms of hypersurface singularities or transverse sections 
of such.  Such discriminants are transverse unions of discriminants of individual 
hypersurface singularities, each of which have the ULT property by a 
result of Saito [Sa].  This continues to hold for transverse sections.  

We shall refer to these as the local components of M̃ in a 
neighborhood of z.
There is then a corollary of the preceding.


Suppose that M̃ is an (n-1)–dimensional Whitney stratified set in 
^n such that: at every smooth point z of M̃, the hypotheses 
of Proposition <ref> holds; and at all singular points M̃ is 
locally the finite union of Whitney stratified sets Y_i each having the ULT 
property.  Then, 


  i) The envelope of M of M̃ has a unique point x ∈ M for 
each z ∈M̃_reg, and M is smooth at all points corresponding 
to 
points in M̃_reg.  

  ii)  At each singular point z of M̃, there is a point in M 
corresponding to each local component of M̃ in a neighborhood of 
z. 



First, if z ∈M̃_reg and satisfies the conditions of Proposition 
<ref>, then there is a unique envelope point corresponding to z 
and the envelope is smooth at that point.  Second, via the isomorphism ν̃ and the commutative diagram 
(3.1), the envelope construction corresponds to the inverse ^∨ of 
 (or rather a local version since we have an orientation).  Under the 
isomorphism ν̃, for each point z ∈M̃_sing there 
corresponds a unique point in the envelope for each local component of 
M̃ containing z. It is obtained as ^∨ applied to the 
unique 
limiting tangent space of z associated to the local component in M̃_reg.



§ INDUCED GEODESIC FLOW BETWEEN HYPERSURFACES

 
We can bring together the results of the previous sections to define the 
Lorentzian geodesic flow between two smooth generic hypersurfaces with 
a correspondence.  We denote our hypersurfaces by M_0 and M_1 and 
let χ : M_0 → M_1 be a diffeomorphism giving the correspondence.  
Note that we allow the hypersurfaces to have boundaries.  We suppose that both are oriented with unit normal vector fields _0 
and _1.  We also need to know that they have a local relative 
orientation.


We say that the oriented manifolds M_0 and M_1, with unit normal 
vector fields _0 and _1, and with correspondence χ : M_0 
→ M_1 are relatively oriented if there is a smooth function 
θ(x) : M_0 → (-π, π) such that _0(x) ·_1(χ (x)) = 
θ(x) for all x ∈ M_0.  

 
An example of a Lorentzian geodesic flow between curves in ^2 is illustrated in Figure <ref>. 
 
If the preceding example in Figure <ref> is slightly perturbed, then the existence 
of a nonsingular Lorentzian flow is guaranteed by the next theorem.
[ Existence, Smooth Dependence and Stability of 
Lorentzian 
Geodesic Flows ]
Suppose smooth generic hypersurfaces M_0 and M_1 are oriented by 
smooth unit normal vector fields _i, i = 0, 1 and are relatively 
oriented by θ for the diffeomorphism χ : M_0 → M_1.  


  * (Existence and Smoothness:)  Then for the given relative 
orientation, is a smooth Lorentzian 
geodesic flow ψ_t : M_0 × [0, 1] →^n between M_0 and 
M_1 given by (<ref>).  
  * (Stability:)  There is a neighborhood  of χ in (M_0, M_1) (for the C^∞–topology) such that if χ^'∈, then M_0 and M_1 are relatively oriented for χ^' 
and the map Ψ : → C^∞(M_0 × [0, 1], ^n) mapping 
χ^' to the associated Lorentzian flow ψ̃^'_t 
is continuous.  
  * (Smooth Dependence:)  Let χ_s : M_0  s→ M_1  s 
be a smooth family of diffeomorphisms between smooth families of 
hypersurfaces for s ∈ S, a smooth manifold (i.e. M_i  s is the 
image of M_i × S under a smooth family of embeddings) so that 
M_0  s and M_1  s are relatively oriented for χ_s for each 
s by a smooth map θ(x, s) : M_0  s→ (-π, π) in (x, s).  Then, 
the family of Lorentzian Geodesic flows 
ψ̃_s, t : M_0 × S × [0, 1] →^n between 
M_0  s and M_1  s is a smooth function of (x, s, t).

  
 Using the form of the Lorentzian geodesic flow given by Proposition 
<ref> we have the Lorentzian geodesic flow is defined by

    ψ_t(x)    =   (t, θ(x))  z_1(x)   +  (1-t, 
    θ(x))  z_0(x)   
           0 ≤ t ≤ 1

Here z_0(x) = (_0(x), c_0(x)) for T_xM_0 defined by _0(x) 
· = c_0(x), and z_1(x) = (_1(x), c_1(x)) for T_χ(x)M_1 
defined by _1(x) · = c_1(x).  As z_i(x) and θ(x) 
depend smoothly on x ∈ M and (t, θ) is smooth on [0, 1] 
× (-π, π), ψ_t(x) is smooth in (x, t).  
Hence, the Lorentzian flow is a smooth well-defined flow between 
(_0(x), c_0(x) ) and (_1(χ(x)), c_1(χ(x)) )  
For smooth dependence 3), we use an analogous argument.  We use 
(<ref>) but with θ(x) replaced by θ(x, s) and each 
z_i(x) by z_i(x, s) = (_1(x, s), c_1(x, s)) where T_xM_0  s is 
defined by _0(x, s) · = c_0(x, s) and T_χ(x, s)M_1  s 
is defined by _1(x, s) · = c_1(x, s).  
Finally to establish the stability, given χ for which M_0 and M_1 
are relatively oriented via the smooth function θ(x), we let 
(x) be a smooth nonvanishing function such that  (x) < 1/3 (π - 
θ(x)) and lim(x) = 0 as x approaches any unbounded 
boundary component at ∞  of M_0.  Then, as (-π, π) is 
contractible there is a Whitney open neighborhood  of χ such that 
if χ^'∈ 
then there is a smooth θ^' : M_0 →  (-π, π) such that 
_0(x) ·_1(χ^'(x)) = cos(θ^'(x)) and 
| θ^'(x) - θ(x)| < (x) for all x ∈ M_0.  Furthermore, 
θ^' depends continuously on χ^'.  
Thus, the corresponding flow in (<ref>) defined by 
θ^' depends continuously on χ^'.  
Specifically, given χ^'∈, consider the mapping 
χ^'_ : M_0 →^n ×^n defined by x ↦ 
((_0(x), c_0(x)), (_1(x), c_1(x))), where (_0(x), c_0(x)) defines 
the tangent space T_xM_0 and (_1(x), c_1(x)) defines the tangent 
space T_χ^'(x)M_1.  Then, χ^'_ is defined 
using the first derivatives of the embeddings M_i ⊂^n and 
χ^' composed with algebraic operations.  Each such operation 
is continous in the Whitney C^∞–topology and so defines a 
continuous map ^' : → C^∞(M_0, ^n ×^n).  Lastly, the Lorentzian flow ψ_t is defined by (<ref>), 
and is the composition of ^' with algebraic operations 
involving the smooth functions 
(x, θ), and is again continuous in the C^∞–topology.  
Hence, the combined composition mapping χ'→ψ_t is 
continuous in the C^∞–topology.  



We note there are two consequences of 2) of Theorem <ref>.  First, 
M_0 and M_1 may remain fixed, but the correspondence χ varies 
in a family.  Then the corresponding Lorentzian geodesic flows vary in a 
family.  Second, M_0 and M_1 may vary in a family with a 
corresponding varying correspondence, then the Lorentzian geodesic flow 
will also vary smoothly in a family.
 


 §.§ Nonsingularity of Level Hypersurfaces of Lorentzian Geodesic Flows in ^n
 It remains to determine when the corresponding Lorentzian geodesic flows 
in ^n will have analogous properties.  We give a criterion involving a 
generalized eigenvalue for a pair of matrices.  We consider the vector fields on M_0, _0(x) and _1(χ(x)).  
For any vector field (x) on M_0 with values in ^n, we let  
N(x) = ((x)  |   d(x)) be the n × n matrix with columns 
(x) viewed as a column vector and d(x) the n × (n-1) 
Jacobian matrix.  If we have a local parametrization (u) of M_0, 
then we may represent the vector field  as a function of u, 
(u).  Then, N((u)) is the n × n matrix with columns 
(u), _u_1(u), … , _u_n-1 (u).  
We denote this matrix for _0 by N_0(x), and that for 
_1(χ(x)) by N_1(x) (or N_0(u) and N_1(χ(u)) if we have 
parametrized M_0).  By i) of Proposition <ref> the Lorentzian 
geodesic flow in ^n will be well-defined provided the corresponding 
matrix N_t(x) is nonsingular at all points of the flow ψ̃_t(x).  We determine this by decomposing N_t(x) into two parts. First, there is the parametrized family of n × 
n–matrices 

    N_t(x)   def=  (t, θ)  N_1(x)   +  (1-t, θ)  N_0(x)

This captures the change resulting from the change in x.  To also capture the change resulting from that in θ, we introduce a second matrix θ _0 whose first column equals the vector 0 and whose j+1–th column is the vector 
θu_j _0, for j = 1, … , n-1.
Then, the nonsingularity criterion will be based on whether the pair of matrices 
(Ñ_t(x), θ _0) does not have a specific 
generalized eigenvalue.  Specifically we introduce one more function.  




    (x, θ)   =   θ(x, θ) - x (θ) (x, θ)

Then, we compute for θ≠ 0

    (x, θ)θ   =   x sin(θ) 
    cos (x θ)  - sin(x θ) cosθ/sin^2θ

and (x, θ)θ_| θ = 0 = 0.
Using (<ref>), a direct calculation shows for all 0 ≤ x ≤ 1.

    (x, θ)   =   cos ((1-x)θ) sin(xθ)   -  x sinθ/sin (xθ) sinθ   =   cos ((1-
    x)θ) /sinθ  -  x/sin (xθ)

if 0 < |θ | < π; and 

    (x, 0)   =     0

We also define 

    N_t^'(x)   def=  Ñ_t(x)    +  (t, 
    θ) θ _0

Then, for any pair (x, t), N_t^'(x) is 
singular  iff -(t, θ(x)) is a generalized  eigenvalue 
for (Ñ_t(x), θ _0). 
Consider the Lorentzian geodesic flow ψ̃_t(x) = 
(_t(x), c_t(x)) between (x) = (_0(x), c_0(x)) and 
(χ(x)) = (_1(χ(x)), c_1(χ(x))) for all x ∈ M_0.  
We let M̃_t = ψ̃_t(M_0), and we let M_t denote the 
envelope of M̃_t.   Then there are the following properties for the envelopes M_t of the 
flow for all time 0 ≤ t ≤ 1.  

Suppose smooth generic hypersurfaces M_0 and M_1 are oriented by 
smooth unit normal vector fields _i, i = 0, 1 and are relatively 
oriented by θ(u).  
Let ψ̃_t be the Lorentzian geodesic flow between M̃_0 
and M̃_1 which is smooth.  If M_t is the family of envelopes 
obtained from the flow M̃_t =  ψ̃_t(M̃_0), then 
suppose that for each time t, M̃_t has only generic Legendrian 
singularities as in <ref> (as e.g. in Fig. <ref>).  Then, 


  * M_t will have a unique point corresponding to z = ψ̃_t(x)  ∈M̃_t provided (<ref>) is nonsingular.  

  * The envelope M_t will be smooth at points corresponding to a 
smooth point z ∈M̃_t satisfying (<ref>) provided 
 H(_t(x))·_t(x) is nonsingular.  Here _t(x) is defined from _t(x) as in <ref>.

  * At points corresponding to singular points z ∈M̃_t, there 
is a unique point on M_t for each local component of M̃ in a 
neighborhood of z.  This point is the unique limit of the envelope points 
corresponding to smooth points of the component of M̃_t 
approaching z.  



We observe that as a result of Theorem <ref>, we can remark about the 
uniqueness of the resulting geodesic flow from non-parabolic points of M_0.  Then, N_0(u) is non singular for each non-parabolic point (u).  If N_1(χ(u)) is 
sufficiently close to N_0(u) then N_t(u) will be nonsingular.  This is 
given by a C^1-condition on the normal vector fields to the surfaces.  If in addition, 
θ(u), the angle between _0(u) and _1(χ(u)), has small variation as a function of u, then the term (t, θ) θ will be small in the C^0 sense.  Thus, if it is sufficiently small, then together with the C^1 closeness of (nonsingular) N_0(u) and N_1(χ(u)) implies that N_t^'(u) is nonsingular.  Hence, by i) of Theorem <ref> the flow is uniqely defined.  Together these are C^2 conditions on N_0(u) and N_1(χ(u)). 











For 2), given that 1) holds, we may apply ii) of Proposition <ref>.  
For 3) we may apply Corollary <ref>.  To prove 1), we will apply i) 
of 
Proposition <ref>.  We must give a sufficient condition that N_t(x) 
is nonsingular for 0 ≤ t ≤ 1.  We choose local coordinates u for a 
neighborhood of _0.  
For a geodesic (_t(u), c_t(u)) between (_0(u), c_0(u)) 
and (_1(u), c_1(u)) given by (<ref>), we must compute 
_t  u_i(u).  We note that not only _i,  i = 1, 2 but also 
θ 
depends on u.  We obtain

    _t  u_i   =   (t, θ) _1  u_i  +  (1-t, 
    θ) _0  u_i  +  (t, θ)u_i _1   +  (1-t, θ)u_i _0

Then, (t, θ)u_i = θu_i (t, 
θ)θ.  Applying (<ref>) with x = t and 1-t, we obtain for the last two terms on the RHS of (<ref>) 
    (t, θ)u_i _1   +  (1-t, θ)u_i _0   
        =   θu_i(  t cos(t θ)/sinθ_1   +  (1 - t) cos((1 - t) θ)/sinθ_0     
          - θ  ((t, θ) _1   +  (1-t, θ) _0 ) )

We see that the last expression in (<ref>) is a multiple of _t.  
We can subtract a multiple of _t from  _t  u_i  without 
altering the rank of the matrix N_t.  Then, after subtracting 
θu_i θ _t from the RHS of (<ref>), 
we obtain

    θu_i(  t cos(t θ)/sinθ_1   + 
     (1 - t) cos((1 - t) θ)/sinθ_0 )

Then, in addition, we can subtract 
θu_i  t  (t θ)  _t from the RHS of 
(<ref>) so the term involving _1 is removed.  We are left with

    θu_i (  (1- t) cos((1- t) θ)/sinθ  -  t   (t θ) sin ((1-t) θ)/sinθ) _0

Adding the two terms in the parentheses in (<ref>), rearranging,  
and using the formula for sin (A + B), we obtain (t, θ), so that 
(<ref>) becomes θu_i  (t, θ) _0.  Thus, 
applying the preceding to each _t  u_i we may replace each of 
them with 

    (t, θ) _1  u_i  +  (1-t, θ) _0  u_i  
    +  θu_i  (t, θ) _0

without changing the rank.  We conclude that N_t has the same rank as 
the matrix N_t^' given in (<ref>).  It remains to consider the case when θ = 0.  Then, both 
θ(t, 0) = 0 and θ(t, 0) = 0, so that the nonsingularity 
reduces to that for Ñ_t(x).

 If _1(χ(x_0)) ≠_0(x_0), then there is a neighborhood x_0 
∈ W ⊂ M_0 such that _1(χ(x)) ≠_0(x) for 
x ∈ W.  Then, there is a smooth unit tangent vector field  defined 
on W such that _1(χ(x)) lies in the vector space spanned by 
_0(x) and (x), and _1(χ(x)) ·(x) ≥ 0 for all 
x ∈ W.  Then, smoothness follows explicitly using the geodesics given 
in Proposition <ref> by (<ref>).










§ RESULTS FOR THE CASE OF SURFACES IN ^3

 
Now we consider the special case of surfaces M_i ⊂^3, i = 1, 2 
for which there is a correspondence given by the diffeomorphism χ : 
M_0 → M_1.  We suppose each M_i is a generic smooth surface with 
_0 = (a_1, a_2, a_3) and _1 = (a^'_1, a^'_2, 
a^'_3)  smooth unit normal vector fields on M_0, respectively 
M_1.  We assume that X(u_1, u_2) is a local parametrization of M_0.  Each 
a_i is a function of  (u_1, u_2) via the local parametrization X(u_1, u_2).  
Likewise, each a_i^' is a function of  (u_1, u_2) via the local 
parametrization χ∘ X(u_1, u_2)
Also, let _i(u) · = c_i(u) define the tangent planes for M_0 
at X(u_1, u_2), respectively  M_1 at χ (X(u_1, u_2)).  We let

    _t    =     (a_1   t, a_2   t, a_3   t)    =   (t, 
    θ)  (a^'_1, a^'_2, a^'_3)   +  (1-t, 
    θ)  (a_1, a_2, a_3)

and c_t(u) = (t, θ)  c_1    +  (1-t, θ)  c_0.  Then,

    N_t    =   [      a_1   t a_1   t, u_1 a_1   t, u_2;      a_2   t a_2   t, u_1 a_2   t, u_2;      a_3   t a_3   t, u_1 a_3   t, u_2 ]

 Note here and what follows we use the following notation.  For quantities 
defined for a flow, we denote dependence on t by a subscript.  We also 
want to denote partial derivatives with respect to the parameters u_i 
by a subscript.  To distinguish them, the subscripts appearing after a comma 
will denote the partial derivatives.  Hence, for example, in (<ref>) 
a_i   t, u_j = a_i   tu_j. 



 §.§ Existence of Envelope Points
  
The sufficient condition that there is a unique point X_t_0(u) in the 
Lorentzian geodesic flow in ^3 at time t = t_0  is that 
(<ref>) 
evaluated at t = t_0 and u = (u_1, u_2) is nonsingular.  
Then, the unique point is the solution of the linear system. 

    N_t_0^T ·   =

with  and   column matrices with entries x_1, x_2, x_3, 
respectively c_t_0, c_t_0, u_1,c_t_0, u_2, 
a_i   t, u_j = a_i   tu_j, and N_t_0 is given by (<ref>). Furthermore, the nonsingularity of (<ref>) is equivalent to that of 
(<ref>).

    N^'_t_0   =   (t_0, θ)   N_1   +  (1- t_0, 
    θ)   N_0  +   (t_0, θ)  θ _0

where 

    θ _0    =   [          0  θ_u_1 a_1  θ_u_2 a_1;          0 θ_u_1  a_2  θ_u_2 a_2;          0  θ_u_1 a_3  θ_u_2 a_3 ]



 §.§ Smoothness of the Envelope
  
For the smoothness of M_t_0 at the point X_t_0(u_1, u_2), we let 

    _t_0 = (a_1   t_0, a_2   t_0, a_3   t_0, - c_t_0)
 
 evaluated at u = (u_1, u_2).   Also, we let _t_0 = _t_0×_t_0  u_1×_t_0  u_1, which is the analogue of the 
cross 
product but for vectors in ^4.  It is the vector whose j–th entry is 
(-1)^j+1 times by taking the 3 × 3 determinant of the 
submatrix 
obtained by deleting the j–th column of 

    [       a_1   t_0       a_2   t_0       a_3   t_0          -c_t_0; a_1   t_0,  u_1  a_2   t_0, u_1  a_3   t_0, u_1     -c_t_0, u_1;  a_1   t_0, u_2  a_2   t_0, u_2  a_3   t_0, u_2    -c_t_0, 
    u_2 ]

Then, we form the 2 × 2–matrix H(_t(u)) ·_t) 
with ij–th entry _t, u_i u_j(u) · _t (u) for u = (u_1, 
u_2).  Then, from Theorem <ref>, we conclude that for a point 
uniquely defined by (<ref>) the envelope is smooth at X_t_0(u) 
if H(_t_0(u))·_t_0(u) is nonsingular.  

 §.§ Envelope Points corresponding to Legendrian Singular 
Points

Third, the generic Legendrian singularities for surfaces are those given in 
Fig. <ref>).  For these:  

  * At points on cuspidal edges or swallowtail points z ∈M̃_t, there is a unique point on M_t which is the unique limit of the envelope 
points corresponding to smooth points of M̃_t approaching z.

  * At points z ∈M̃_t which are tranverse intersections of 
two or three smooth surfaces, or the transverse intersection of a smooth surface
 and a cuspidal edge, there is a unique point in M_t for each smooth surface 
passing through z (and one for the cuspidal edge).  

 

As an example, we consider the Lorentzian geodesic flows between the surfaces M_1 given by z = 2 - .2 (x^2 + y^2), M_2 given by z = .5 - .05 (x^2 + y^2), 
and M_3 given by z = 4 - .5 (x^2 + y^2).  We consider two correspondences 
and the resulting Lorentzian geodesic flow between them.  The first assigns to 
each point in M_1 the point in M_3 with the same coordinates (x, y) so the points on the same vertical lines correspond.  For the second, each point in M_2 corresponds to the point in M_3 in the same vertical line.  For the third, we assign to each point (x, y, z) of M_2 the point (2/5 x, 2/5 y, 2/5 z + 3.2) in M_3. Although for the first two there are simple Euclidean geodesic flows in ^3 along the vertical lines, these are not the Lorentzian geodesic flow lines.  
 
 
For the third, M_3 is obtained from M_2 by a combination of the homothety of multiplication by 2/5 combined with the translation by (0, 0, 3.2).  Hence, for the second case the Lorentzian geodesic flow is given by Corollary <ref> to be along the lines joining the corresponding points and is given by 

    (x, y, z)  ↦  ((1- 3/5 t) x, (1- 3/5 t) y, (1- 3/5 t) z + 3.2 t)   .
 By the circular symmetry of each surface about the z-axis, we may view the 
Lorentzian geodesics in a vertical plane through the z-axis.  We may compute both the level sets of the Lorentzian geodesic flow and the corresponding geodesics 
using Proposition <ref> and solving the systems of equations (<ref>).  We show the results of the computations using the software Maple in Figures <ref> and <ref>.  The Lorentzian geodesic flow between M_1 and M_3 with the vertical correspondence is nonsingular, as shown by the level sets and geodesic curves in Figure <ref>.  
By comparison, the Lorentzian geodesic flow between M_2 and M_3 for the vertical correspondence is singular.  We see the cusp formation in the level sets in Figure <ref> a).  The singularities result from the intersection of the geodesics seen in  and the individual flow curves in b).  We also see that the increased bending of the 
geodesics versus those in Figure <ref> result from the increases in the changes in tangent directions, leading to the formation of cusp singularities.  By contrast, for the second correspondence resulting from the action of the element of the extended Poincare group, geodesics are straight lines as shown in Figure <ref> and the flow is 
nonsingular.  
 
 For the surfaces, the flows are obtained by rotating the planar figures in Figures <ref>, <ref> and <ref>.  The rotation of Figure <ref> yields a circular cusp edge which evolves from a single point on the axis of symmetry.  Hence, the creation point for the cusp singularities does not have generic form. 

 

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