Order of contact and ruled submanifolds
Igor Uljarevic

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    [
    December 30, 2023
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We prove a generalization of the Monge-Cayley-Salmon theorem on osculation and ruled submanifolds using geometric measure theory.




§ INTRODUCTION


Analytic surfaces in ^3 have the following remarkable property, that played a key role in the proof of the Erdős distinct distances problem in dimension two by Guth and Katz <cit.>.


Let M⊂^3 be a proper 3-dimensional analytic surface. Assume there exists a smooth family ℓ_x,x∈ M of lines in ^3 such that, for all x∈ M, ℓ_x and M have a contact of order 3 at x. Then, ℓ_x⊂ M for all x∈ M.


A proof of Theorem <ref> can be found in <cit.>. In <cit.>, Guth and Zahl proved a version of Theorem <ref> for an arbitrary field instead of . For a detailed exposition on the Monge-Cayley-Salmon theorem and its relation to the Erdős distinct distances problem, we refer to <cit.> and <cit.>. The aim of the present paper is to prove the following generalization of Theorem <ref>. Along the line, we present  a novel elementary proof of the Monge-Cayley-Salmon theorem.


A curve Γ⊂^n is said to be of class k∈ℕ if it can be parametrized by a map →^n whose coordinates are polynomial functions of degree at most k.


In the terminology of Definition <ref>, the lines are curves of class 1.


Let M⊂^n be a proper m-dimensional analytic submanifold. Assume there exists a smooth family Γ_x,x∈ M of class-k curves in ^n such that, for all x∈ M, Γ_x and M have a contact of order k(m+1) at x. Then, Γ_x⊂ M for all x∈ M.


The proof of Theorem <ref> for k=1 and for analytic submanifolds of ℂ^n or ℂP^n can be found in <cit.>. That proof uses techniques of algebraic geometry. 

The main idea of our proof is to consider the (m+1)-dimensional volume swept by M as each point x of M moves along Γ_x. It turns out (see Proposition <ref> on page prop:contained) that this volume is equal to 0 precisely when Γ_x⊂ M for all x∈ M. The order-of-contact condition, on the other hand, implies that the rate at which the volume is swept is sufficiently slow (see Proposition <ref> on page prop:analyticlimit0). What bridges these two facts (the vanishing volume and the volume being swept at a sufficiently slow rate) is a result of a Weyl-tube-formula type (Proposition <ref> on page prop:notarbitrary). Now, we sketch this step in the case of a hypersurface in ^n. The volume swept by the hypersurface is a polynomial in the time-variable t of degree at most k(m+1)=kn. If a polynomial of degree at most kn grows slower than t^kn as we approach 0, then it vanishes identically. 



 §.§ Acknowledgments


I would like to thank Filip Morić for suggesting this project. I am grateful to Darko Milinković for many useful discussions. This work was supported by the Ministry of Education, Science, and Technological development, grant number 174034.



§ PRELIMINARIES




 §.§ The nearest point map



Let M⊂^n be a smooth submanifold, and let N be a normal tubular neighbourhood of M. The nearest point map r:N→ M is the map that sends each point of N to the unique nearest point in M.


The nearest point map of a smooth submanifold is smooth <cit.>. If the submanifold is analytic, then the nearest point map is analytic as well <cit.> .



 §.§ Order of contact



Two smooth curves γ_1,γ_2:→^n are said to have a contact of order k∈ℕ∪{0} at a point t_0∈ if

    (∀ j∈{0,1,…,k})  γ_1^(j)(t_0)=γ_2^(j)(t_0).




A smooth curve γ_1:→^n is said to have a contact of order k∈ℕ∪{0} with a smooth submanifold M⊂^n at a point t_0∈ if there exists a smooth curve γ_2:→ M such that the curves γ_1 and γ_2 have a contact of order k at t_0.



Two submanifolds M_1,M_2⊂^n are said to have a contact of order k∈ℕ∪{0} at a point p∈ M_1∩ M_2 if for every smooth curve γ_1:→ M_1 such that γ_1(0)=p there exists a smooth curve γ_2:→ M_2 such that γ_1 and γ_2 have a contact of order k at 0.


In the following lemma (and in the rest of the paper), we denote by d(x, M) the distance between a point x∈^n and a subset M⊂^n, i.e.

    d(x,M):=inf{x-y  |   y∈ M }.



Let M be a submanifold of ^n, let I be an open interval containing 0, and let ϕ_t:M→^n, t∈ I be a smooth family of embeddings. Assume, for all x∈ M, the curve t↦ϕ_t(x) has a contact of order k∈ℕ∪{0} with M at t=0. Then,

    lim_t→ 0d(ϕ_t(x),M)/t^k=0

uniformly on compact subsets of M.


Let N be a normal tubular neighbourhood of M, and let r:N→ M be the nearest point map. Let K⊂ M be an arbitrary compact subset, and let δ>0 be such that [-δ,δ]⊂ I and such that ϕ_t(x)∈ N for x∈ K and t∈[-δ,δ]. Denote

    C:=max_x∈ K, t⩽δ∂_t^k+1(ϕ_t(x)-r∘ϕ_t(x)).

Fix x∈ M, and denote by γ=(γ_1,…,γ_n): I→^n the curve defined by γ(t)=ϕ_t(x)-r(ϕ_t(x)). Since ϕ_t(x) has a contact of order k with M, the derivatives of γ up to order k are equal to 0. The Taylor approximation implies

    γ_i(t)⩽t^k+1/(k+1)!·max_t⩽δ∂_t^k+1γ_i(t)⩽ t^k+1·C/(k+1)!,

for all i∈{1,…, n} and t∈(-δ,δ). Hence

    d(ϕ_t(x),M)/t^k=γ(t)/t^k⩽ t·√(n)· C/(k+1)!,

for all x∈ K and t∈ (-δ,δ). This finishes the proof.




 §.§ Volume of a C^1 map


The purpose of this section is to introduce the notion of a volume Vol(ϕ) of a C^1 map ϕ:M→ N from a smooth manifold M to a Riemannian manifold (N,g). Intuitively, the volume of a C^1 map ϕ:M→ N is the ( M)-dimensional volume swept by ϕ in N. We will, actually, formally define only the volume of a C^1 map form an open subset of ^m to a Riemannian manifold. This definition extends to the general case via the standard trick which uses a collection of charts and a subordinate partition of unity.


Let U⊂^m be an open subset, let (N,g) be a Riemannian manifold, and let ϕ: U→ N be a C^1 map. The volume of the map ϕ is defined by

    Vol(ϕ):=∫_U∂_1ϕ(x) ∧⋯∧∂_mϕ(x) dx.




We found it convenient to use exterior algebra ⋀^m T_ϕ(x)N to express the volume element. In our conventions, if e_1,…, e_n is an orthonormal basis of T_ϕ(x)N, then e_i_1∧⋯∧ e_i_m, 1⩽ i_1<⋯<i_m⩽ n is an orthonormal basis of ⋀^m T_ϕ(x)N. Alternatively,   ∂_1ϕ(x) ∧⋯∧∂_mϕ(x) can be written as

    ∂_1ϕ(x) ∧⋯∧∂_mϕ(x) = √([ g(∂_iϕ(x), ∂_jϕ(x)) ]_i,j).




Let ψ:V→ U be a C^1 diffeomorphism between two m-dimensional manifolds, let (N,g) be a Riemannian manifold, and let ϕ:U→ N be a C^1 map. Then, Vol(ϕ∘ψ)=Vol(ϕ). 


Without loss of generality, assume U and V are two open subsets of ^m. For a linear map A:W_1→ W_2 (from a vector space W_1 to a vector space W_2), denote by ⋀ ^k A :⋀^k W_1→⋀^k W_2 the linear map defined by

    (∀ v_1,… v_k∈ W_1)  (⋀^k A)(v_1∧⋯∧ v_m):=(A v_1)∧⋯∧ (A v_k).

The lemma follows from the following sequence of equalities

    Vol(ϕ∘ψ)   = ∫_V∂_1(ϕ∘ψ) ∧⋯∧∂_m(ϕ∘ψ) dx
       = ∫_V  (Dϕ(ψ(x))∂_1ψ(x)) ∧⋯∧ (Dϕ(ψ(x))∂_mψ(x)) dx
       = ∫_V ⋀^k(Dϕ(ψ(x))) ∂_1ψ(x) ∧⋯∧∂_m ψ(x) dx
       = ∫_V ⋀^k(Dϕ(ψ(x))) Dψ(x) e_1 ∧⋯∧ e_m  dx
       = ∫_V ⋀^k(Dϕ(ψ(x))) e_1 ∧⋯∧ e_m ·Dψ(x)dx
       = ∫_V Dϕ(ψ(x))e_1 ∧⋯∧ Dϕ(ψ(x))e_m ·Dψ(x)dx
       = ∫_V ∂_1ϕ(ψ(x)) ∧⋯∧∂_mϕ(ψ(x))·Dψ(x)dx
       = ∫_U ∂_1ϕ(y) ∧⋯∧∂_mϕ(y)dy
       = Vol(ϕ).

Here, e_1,…, e_m stands for the standard basis of ^m.




§ FAMILY OF EMBEDDINGS AND THE SWEPT VOLUME


In this section, we consider the maps of the form

    ϕ: M×→^n,
       ϕ(x,t):= x+ t· v_1(x)+⋯ t^k· v_k(x),

where M is an m-dimensional submanifold, and v_i:M→^n is a smooth map for i∈{1,…,k}. Proposition <ref> proves that the rate of growth of Vol(.ϕ|_M×(-t,t)) at 0 cannot be arbitrary. More precisely, it shows that Vol(.ϕ|_M×(-t,t))=o(t^k(m+1)) implies Vol(.ϕ|_M×(-t,t))≡ 0. Proposition <ref> is a general statement about a smooth 1-parameter family of embeddings with vanishing volume. Proposition <ref> relates the order of contact with the growth rate of Vol(.ϕ|_M×(-t,t)) at 0. In this proposition, it is assumed that M is an analytic submanifold of ^n.


Let M⊂^n be an m-dimensional submanifold, and let v_i:M→^n,  i∈{1,…,k} be smooth maps. Denote by ϕ:M×→^n the map defined by

    ϕ(x,t):=x+t· v_1(x)+⋯+ t^k· v_k(x).

If

    lim_t→ 0Vol(.ϕ|_M×(-t,t))/t^k(m+1)=0,

then Vol(ϕ)=0.


Denote d:= k(m+1)-1. Let α: U →^m, where U⊂^m is open, be a parametrization of a subset of M. Denote by ψ: U×→^n the map defined by ψ(x,t):=ϕ(α(x),t). The map ψ is equal to the composition of the restriction .ϕ|_α(U)× with the diffeomorphism

    U×→α(U)×  :   (x,t)↦ (α(x),t).

Hence

    Vol(.ψ|_U×(-t,t))=Vol(.ϕ|_α(U)×(-t,t))⩽ Vol(.ϕ|_M×(-t,t)),

and, consequently, 

    lim_t→ 0Vol(.ψ|_U×(-t,t))/t^d+1=0.

The volume element ∂_1ψ(x,t)∧⋯∧∂_mψ(x,t)∧∂_tψ(x,t) is equal to

    ⋀_i=1^m( ∂_iα(x) + ∑_j=1^k t^j·∂_i (v_j∘α)(x))∧(∑_j=1^k j· t^j-1· (v_j∘α)(x)).

After developing the expression above by distributive law and after applying the Pythagorean theorem, the volume element transforms into the form

    √(A_1(x,t)^2+⋯+A_ℓ(x,t)^2)

where A_1,…, A_ℓ are polynomials in t of degree at most d whose coefficients are smooth functions U→ in x-variable, and ℓ:=nm+1. Assume there exists j∈{1,…, ℓ} such that A_j(x,t) is not identically equal to 0. Then,

    A_j(x,t)=a_0(x)+a_1(x)· t+⋯+ a_d(x)· t^d,

where a_1,…,a_d:U→ are smooth functions that are not all identically equal to 0. Let b∈{0,…,d} be the smallest index such that a_b is not identically equal to 0. There exists ε>0 such that

    1/2∫_Ua_b(x)dx·t^b⩾∑_i=b+1^d ∫_U a_i(x)dx ·t^i,

for t<ε. Hence, for t<ε,

    ∫_U√(A_1(x,t)^2+⋯+A_ℓ(x,t)^2) dx
    [origin=c]-90⩾
    ∫_UA_j(x,t)dx
    [origin=c]-90⩾
    ∫_Ua_b(x)dx·t^b- ∑_i=b+1^d∫_Ua_i(x)dx·t^i
    [origin=c]-90⩾
    1/2∫_Ua_b(x)dx·t^b

This further implies

    0   =lim_t→ 0Vol(.ψ|_U×(-t,t))/t^d+1
       ⩾lim_t→ 0t^-(d+1)·∫_-t^t 1/2∫_Ua_b(x)dx·s^bds
       = lim_t→ 0t^-(d+1)·∫_0^t ∫_Ua_b(x)dx·s^bds
       =lim_t→ 0 t^-(d+1)·t^d+1/d+1·∫_Ua_d(x)dx
       = 1/d+1∫_Ua_b(x)dx.

The continuity of a_b now implies a_b(x)=0 for all x∈ U. Contradiction! Therefore 

    A_1(x,t)=⋯=A_ℓ(x,t)=0,

for all x∈ U and t∈, and, consequently, 

    Vol(.ϕ|_α(U)×)=Vol(ψ)=0.

This holds for all charts α of M. Hence Vol(ϕ)=0.



Let M be an m-dimensional manifold, let I be an open interval containing 0, and let ϕ_t:M→^n be a smooth family of embeddings. Assume Vol(ϕ: M× I→^n)=0. Then, for all x∈ M, there exists ε=ε(x)>0 such that 

    (∀ t∈ (-ε,ε))  ϕ_t(x)∈ϕ_0(M).



 Suppose there exists (x_0,t_0)∈ M× I such that Dϕ(x_0,t_0) is of rank m+1. Then, there exists a neighbourhood U⊂ M× I of (x_0,t_0) such that .ϕ|_U is an embedding. The volume of an embedding is positive. Hence
 
    Vol(ϕ)⩾ Vol(.ϕ|_U)>0.

 This contradicts Volϕ=0. Therefore 

    (∀ (x,t)∈ M× I)  rank Dϕ(x,t)⩽ m.

Since ϕ_t is a family of embeddings, the rank of Dϕ(x,t) is less than m+1 if, and only if, 

    ∂_tϕ_t(x)∈ Dϕ_t(x).

Denote by Y_t the smooth vector field on M defined by 

    ∂_tϕ_t(x)= Dϕ_t(x) Y_t(x).

Denote by ψ the (locally defined) flow of the vector field -Y_t. Fix x∈ M. Let O be a neighbourhood of x and let ε>0 be such that ψ_t(y) is well defined for y∈ O and t∈(-ε,ε).  Let δ>0 be such that ψ_t^-1(x)∈ O for all t∈ (-δ,δ). Since

    ∂/∂ t(ϕ_t(ψ_t(x)))   = Dϕ_t(ψ_t(x))Y_t(ψ_t(x))+ Dϕ_t(ψ_t(x))∂_tψ_t(x)
       = Dϕ_t(ψ_t(x))Y_t(ψ_t(x))+ Dϕ_t(ψ_t(x))(-Y_t(ψ_t(x)))
       =0,

for all y∈ O and t∈(-ε, ε), we have ϕ_t(ψ_t(y))=ϕ_0(y).  By substituting y=ψ_t^-1(x), one gets

    (∀ t∈ (-δ,δ))  ϕ_t(x)=ϕ_0(ψ_t^-1(x)))∈ϕ_0(M).




Let M⊂^n be an analytic submanifold, and let ϕ:M×→^n be a smooth map such that

    
  * ϕ(x,0)=x, for all x∈ M,
    
  * ϕ(x,t)=x, for t∈ and for x outside of a compact set,
    
  * the curve t↦ϕ(x,t) is analytic and has a contact of order k∈ℕ with M at t=0 for all x∈ M.

Then,

    lim_t→ 0Vol(.ϕ|_M×(-t,t))/t^k=0.



Without loss of generality, assume that M is covered by a single chart α:U⊂^m→ M. Let N be a normal tubular neighbourhood of M, and let r:N→ M be the nearest point map. Since ϕ_t(x) is t-independent for x outside of a compact set (and since ϕ_0(x)∈ M), there exists δ>0 such that ϕ(M× (-δ,δ))⊂ N. Let δ_1∈ (0,δ) be such that ϕ_t: M→^n is an embedding for  all t∈ [-δ_1,δ_1]. Such δ_1 exists because the set of embeddings M→^n is open <cit.>. Let Y_t, t∈[-δ_1,δ_1] be the vector field on M defined by

    Dϕ_t(x) Y_t(x)=-dr(∂_tϕ_t(x)).

Denote by θ_t:M→ M the flow of the vector field Y_t. Let ψ_t:M→^n be the smooth family of embeddings defined by ψ_t(x):=ϕ_t(θ_t(x)).  For t∈[-δ_1,δ_1], the following holds

    ϕ_t(M)=ψ_t(M),   Vol(.ϕ|_M×(-t,t))=Vol(.ψ|_M×(-t,t)).

There exists a compact set K⊂ U such that ψ_t(x)=x for x∈α(U∖ K) and t∈[-δ_1,δ_1]. Denote

    C:=max _x∈ K, t⩽δ_1∂_1(ψ_t∘α)(x)∧⋯∧∂_m(ψ_t∘α)(x).

Since

    v_1∧⋯∧ v_ℓ⩽v_1·v_2∧⋯∧ v_ℓ,

we get

    Vol   (.ϕ|_M×(-t,t))=Vol(.ψ|_M×(-t,t))
       =∫_U∫_-t^t∂_1(ψ_s∘α)(x)∧⋯∧∂_m(ψ_s∘α)(x)∧∂_sψ_s∘α(x)dsdx
       ⩽∫_U∫_-t^t∂_sψ_s∘α(x)·∂_1(ψ_s∘α)(x)∧⋯∧∂_m(ψ_s∘α)(x)dsdx
       =∫_K∫_-t^t∂_sψ_s∘α(x)·∂_1(ψ_s∘α)(x)∧⋯∧∂_m(ψ_s∘α)(x)dsdx
       ⩽ C·∫_K∫_-t^t∂_sψ_s∘α(x)ds,

for t∈[0,δ_1). It is enough to prove 

    lim_t→ 01/t^k·∫_-t^t∂_sψ_s(x)ds=0.

Since

    ∂_tψ_t(x)   = (∂_tϕ_t)(θ_t(x))+ Dϕ_t(θ_t(x))∂_tθ_t(x)
       =(∂_tϕ_t)(θ_t(x))+ Dϕ_t(θ_t(x))Y_t(θ_t(x))
       =(∂_tϕ_t)(θ_t(x))- dr(∂_tϕ_t(θ_t(x)))
       ∈ dr,

we have 

    ∂_t r(ψ_t(x)) = dr(ψ_t(x))∂_tψ_t(x)=0,

and, consequently, r(ψ_t(x))=x for t∈(-δ_1,δ_1). By Lemma <ref> below, for x∈ M, there exists ε_x>0 such that the coordinates of r(ψ_t(x))-ψ_t(x)=x-ψ_t(x) are monotone (with respect to t) for t∈ [-ε_x,0] and for t∈ [0,ε_x]. Lemma <ref> below implies

    ∫_-t^t ∂_sψ_s(x)ds   = ∫_-t^t ∂_s(x-ψ_s(x))ds
       =∫_-t^0 ∂_s(x-ψ_s(x))ds + ∫_0^t ∂_s(x-ψ_s(x))ds
       ⩽ n·( ψ_-t(x)-x + ψ_t(x)-x)
       = n·( d(ψ_-t(x), M) + d(ψ_t(x), M)  )
       ⩽ n·( sup_y∈ Md(ψ_-t(y), M) + sup_y∈ Md(ψ_t(y), M) )
       =n·( sup_y∈ Md(ϕ_-t(y), M) + sup_y∈ Md(ϕ_t(y), M) )

for t∈[0,ε_x]. Hence (by Lemma <ref>) (<ref>) holds, and the proof is finished.



Let M⊂^n be an analytic submanifold, and let γ:→^n be an analytic curve such that γ(0)∈ M. Denote by r:N→ M the nearest point map defined in a normal tubular neighbourhood N of M. Then, there exists ε>0 such that the coordinates of the function

    [-ε,ε]→^n  :   t↦ r(γ(t))-γ(t)

are monotone (not necessarily strictly) on [-ε,0] and [0,ε].


Since M is an analytic submanifold of ^n, the nearest point map is analytic <cit.>. The set γ^-1(N) is open. Hence there exists δ>0 such that γ(t)∈ N for all t∈(-δ,δ). Denote by

    f=(f_1,…, f_n):(-δ,δ)→^n

the analytic map defined by f(t)=r(γ(t))-γ(t). Fix j∈{1,…,n}. If f_j^(k)(0)=0 for all k∈ℕ, then (since f_j is analytic) there exists ε>0 such that f_j(t)=0 for t∈(-ε,ε). Consequently, f_j is monotone on (-ε,ε).
Assume, now, there exists k∈ℕ such that f_j^(k)(0)≠0, and such that f_j^(i)(0)=0 for all i∈{1,…,k-1}. The Taylor approximation for f'_j implies

    f_j'(t)    = f_j^(k)(0)/(k-1)!· t^k-1 + f_j^(k+1)(c_t)/k!· t^k
       =t^k-1/(k-1)!( f_j^(k)(0)+1/k· f_j^(k+1)(c_t)· t ),
 
for t∈ (-δ,δ) and for some c_t between 0 and t. Since 1/k· f_j^(k+1) is a bounded function on [-δ/2,δ/2], there exists ε∈(0,δ) such that the function f'_j does not change the sign on intervals (-ε,0) and (0,ε). Hence f_j is monotone on [-ε,0] and [0,ε].



Let [a,b]⊂ be a compact interval, and let γ=(γ_1,…,γ_n):[a,b]→^n be a C^1 curve such that γ_i:[a,b]→ is monotone for all i∈{1,…,n}. Then,

    length(γ)⩽ n·γ(a)-γ(b).



Since

    √(∑_j(γ_j'(t))^2)⩽√(n)·max_jγ_j'(t)⩽√(n)·∑_jγ_j'(t),

the length of γ is bounded by

    length(γ)   ⩽√(n)·∑_j∫_a^bγ_j'(t)dt
       =√(n)·∑_j∫_a^b γ_j'(t)dt
       =√(n)·∑_jγ_j(b)-γ_j(a)
       ⩽ n^3/2·√(1/n·∑_jγ_j(b)-γ_j(a)^2)
       =n·γ(b)-γ(a).

In the sequence of inequalities above, we used 

    ∫_a^bγ'_j(t)dt=∫_a^bγ'_j(t)dt

(which holds because γ_j is monotone) and the Cauchy-Schwarz inequality.




§ PROOF OF THE MAIN THEOREM



Let p be an arbitrary point in M, and let v_1,…, v_k:M→^n be smooth compactly supported maps such that

    x + t· v_1(x) + ⋯ + t^k· v_k(x)∈Γ_x

for all x∈ M, and such that

    t↦ p + t· v_1(p) + ⋯ + t^k· v_k(p)

is a parametrization of Γ_p. Denote by ϕ_t:M→^n, t∈ the family of smooth maps defined by

    ϕ_t(x):=x + t· v_1(x) + ⋯ + t^k· v_k(x).

Proposition <ref> implies

    lim_t→ 0Vol(.ϕ|_M×(-t,t))/t^k(m+1)=0.

There exists ε>0, such that ϕ_t,t∈ (-ε,ε) is a smooth family of embeddings (see <cit.>). Therefore, due to Proposition <ref>, Proposition <ref>, and (<ref>), ϕ_t(x)∈ M for t small enough. In particular, there exists an open segment I_p  of Γ_p such that p∈ I_p⊂ M. Since M is proper, and  since M and Γ_p are analytic, the identity theorem for analytic functions <cit.> implies Γ_p⊂ M.




