On the geometry of some Equivariantly related manifolds
Llohann D. Sperança Leonardo F. Cavenaghi

	
	
		We provide a topological procedure to obtain geometric realizations of both classical and `exotic' G-manifolds, such as spheres, bundles over spheres and Kervaire manifolds. As an application, we apply the process known as Cheeger deformations to produce new metrics of both positive Ricci and almost non-negative curvature on such objects.
	
	The first author is financially supported by CNPq, grant number 131875/2016-7. The second author was financially supported by FAPESP, grant numbers  2009/07953-8 and 2012/25409-6 and CNPq 404266/2016-9.
	
	Towards non-invasive cancer diagnostics and treatment based on electromagnetic fields, optomechanics and microtubules
    J. A. Tuszynski
    December 30, 2023
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§ INTRODUCTION

	
	Compared with other curvature conditions, examples of manifolds with positive sectional curvature are still sparse in literature (see <cit.> for a survey). For instance, there are still no obstructions that distinguish the class of simply connected compact positively curved manifolds from   non-negatively curved ones. In parallel to this situation, it is still not known the existence of metrics of positive sectional curvature in interesting manifolds, such as exotic spheres and sphere bundles over spheres. 
	
	Easier to tackle is the existence of metrics of positive Ricci and positive scalar curvatures. Many geometric/topological constructions are well succeed in this aim. For instance, the topological constructions  in <cit.> and the constructions based on symmetries  in <cit.>.
	
	
	
	
	
	
	Here we provide a common ground for <cit.>, geometric realizing `exotic' manifolds as isometric quotients of principal bundles over `standard' ones. The realization is incarnated in the form of a cross-diagram:
	
    G@..[d]^∙   
     G@..[r]^⋆    P[d]^π[r]^π'   M'
       M

	Special properties of the constructed diagrams allow one to compare the geometries of M and M' (see section <ref> and <ref>). Former instances of diagram (<ref>) are found in <cit.>, mainly exploiting the differential topology of exotic spheres. 
	
	On the other hand, the construction explicitly realizes Morita equivalences between related transformation groupoids[The authors thank O. Brahic and C. Ortiz for pointing it out] which might be of independent interest.
	
	As an application, we establish the existence of metrics of positive Ricci curvature and almost non-negative curvature in new examples (to the best of our knowledge). The construction works specially realizing some connected sums (see section <ref> or Theorem <ref>).
	
	We recall that a manifold M has almost non-negative sectional curvature if it admits a sequence of Riemannian metrics {g_n}_n∈ N satisfying _g_n≥ -1/n and diam_g_n≥ 1/n (see <cit.> for recent developments). 
	
	
		Let Σ^7, Σ^8 be any homotopy sphere in dimensions 7 and 8, Σ^10 any homotopy 10-sphere which bounds a spin manifold and Σ^4m+1 the Kervaire sphere in dimension 4m+1. Then the manifolds below admits a sequence {g_n}_n∈ N of metrics with positive Ricci curvature satisfying _g_n≥ -1/n and diam_g_n≥ 1/n:
		
			
  * M^7#Σ^7, where M^7 is any 3-sphere bundle over S^4, 
			
  * M^8#Σ^8, where M^8 is either a 3-sphere bundle over S^5 or a 4-sphere bundle over S^4
			
  * M^10#Σ^10, where 
			
				
  * (M^8× S^2),  M^8  as in item (2)
				
  * M^10 is any 3-sphere bundle over S^7, 5-sphere bundle over S^5 or 6-sphere bundle over S^4				
			
			
  * M^4m+1#Σ^4m+1 where  
			
				
  * S^2m⋯ M^4m+1→ S^2m+1 is a sphere bundle representing any multiple of the unitary tangent of S^2m+1
				
  * CP^m⋯ M^4m+1→ S^2m+1 is the CP^m-bundle associated to any  multiple of U(m)⋯ U(m+1)→ S^2m+1
				
  * U(m+2)/SU(2)× U(m)	
			
			
  * (M^8m+k× N^5-k)#Σ^8m+5 where N^5-k is any manifold with positive Ricci curvature and
			
				
  * S^4m+k⋯ M^4m+k→ S^4m+1 is a sphere bundle representing the k-th (fiberwise) suspension of any multiple of the unitary tangent of S^4m+1
				
  * k=1, HP^m⋯ M^8m+1→ S^4m+1 is the HP^m-bundle associated to any  multiple of Sp(m)⋯ Sp(m+1)→ S^4m+1
				
  * k=0, M=Sp(m+2)/Sp(2)× Sp(m)
				
  * k=1, M=M^8m+1 as in item  (<ref>)
			
			
	 
	
	
	
	We recall that a homotopy n-sphere is a manifold homotopy equivalent to the unitary sphere of R^n+1. For n>4, homotopy spheres are homeomorphic to standard spheres (Smale <cit.>). It is exotic if it is not diffeomorphic to the standard sphere. As computed by Kervaire and Milnor <cit.>, there are (up to orientation) exactly 14 exotic spheres in dimension 7, one in dimension 8 and one in dimension 10 which bounds a Spin manifold (there are two more that does not). Kervaire spheres are the boundary of P^2n(𝐀_2) (in Bredon <cit.> notation), the plumbing of two copies on the (disc bundle associated to the) tangent bundle of S^n, n=2m+1 (see section <ref> for a more detailed description). The Kervaire sphere is known to be exotic for all but finitely many (known) values of m (see Hill–Hopkins–Ravenel <cit.> and Wang–Xu <cit.>). When n is even, the resulting boundary, Σ^4m-1=∂ P^4m(𝐀_2), has H_2m(Σ^4m-1)≅ Z_3.
	
	As an extra contribution, we provide metrics of positive Ricci curvature on quotients of Σ^4m-1. Whenever n is even or odd, ∂ P^2n(𝐀_2) posses a (bi-axial) O(n)-action which restricts to a free SO(2) (respectively, SU(2)) action when n is even (respectively, when n=0 4). We denote the quotient by Σ P C^n (respectively, Σ P H^n/2). 
	Σ P C^n (respectively, Σ P H^n/2) is an U(n)-manifold (respectively, a Sp(n/2)-manifold).
	
	
		Σ P C^n and Σ P H^n/2 posses invariant metrics with positive Ricci curvature.
	
	
	
	The restriction/variety of examples in Theorem <ref> is related to the presence of symmetries and explicit realizations of spheres. To give a more precise statement, we fix the representations below:
	
	
		
  (ρ_7) n=7, G=S^3: ρ=ρ_1⊕ρ_1⊕ρ_0, where ρ_1:S^3→ O(3) is the  representation induced by the double cover S^3→ SO(3) and ρ_0:S^3→ SO(1) is the trivial one 
		
  (ρ_8) n=8, G=S^3: ρ=ρ_⊕ρ_1⊕ρ_0, where ρ_:S^3→ O(4) is the representation induced by right (or left) multiplication by a quaternion
		
  (ρ_10) n=10, G=S^3: ρ=ρ_⊕ρ_1⊕ 3ρ_0 or ρ=ρ_1⊕ρ_1⊕ρ_1⊕ρ_0
		
  (ρ_4m+1) n=4m+1, G=U(m): ρ=ρ_U(m)⊕ρ_U(m)⊕ρ_0 where ρ_U(m):U(m)→ O(2m) is the standard representation
		
  (ρ_8m+5) n=8m+5, G=Sp(m): ρ=ρ_Sp(m)⊕ρ_Sp(m)⊕ 5ρ_0 where ρ_Sp(m):Sp(m)→ O(4m) is the standard representation
		
	
	
	
	Given a G-invariant metric on M, we recall that M/G is a metric space with orbit distance. The G-orbits have constant dimension on an open and dense set of M^*, called the principal part (see Bredon <cit.>). The subset M^*/G has a natural manifold structure that makes the quotient M^*→ M^*/G a Riemannian submersion.
	
	
		Let M^n be a G-manifold that posses a fixed point whose isotropy representation is (ρ_n). Then: 
		
			
  * if the orbits on M have finite fundamental group and M posses a G-invariant metric such that  Ricci_M^*/G≥1, then M^n#Σ^n has a metric with positive Ricci curvature 
			
  * if  M posses a family of G-invariant metrics such that M^*/G has almost non-negative curvature, then M^n#Σ^n has almost non-negative curvature
			
  * if M posses a family of metrics satisfying (1) and (2), then M^n#Σ^n admits a family of metrics {g_n}_n∈ N with positive Ricci curvature satisfying _g_n≥ -1/n and diam_g_n≥ 1/n 
		
	
	
	
	
	For Theorem <ref>, we construct a diagram as in (<ref>) with G compact and M' diffeomorphic to M^n#Σ^n. For G compact, we prove that, given a G-invariant metric on M, there is a G-invariant metric on M'  such that M'/G is isometric to M/G (see section <ref> for the G-action on M'). Thus, if M satisfies the hypothesis in Theorems A, B or C in Searle–Wilhelm <cit.>, so does M'. Using diagram (<ref>), we also provide a self-contained proof of Theorem <ref> (the greater machinery in <cit.> can be avoided in Theorem <ref> since diagram (<ref>) gives a better control of the geometry on the non-principal part of M).
	
	Some prior works related to Theorem <ref>: the works of Nash <cit.> and Poor <cit.> provides metrics of positive Ricci curvature on sphere bundles over spheres. The procedure in Fukaya–Yamaguchi <cit.> is consistent with Nash <cit.> and Poor <cit.>, thus, providing metrics which are simultaneously Ricci positive and almost non-negative on sphere bundles over spheres. Crowley–Wraith <cit.> considers positive Ricci curvature on connected sums of highly connected odd-dimension manifolds with exotic spheres, covering the Ricci curvature in example (i) and (possibly partially) in example (iv). Almost non-negative curvature in homotopy 7-spheres was established by Searle–Wilhelm <cit.>.
	
	
	The connected sum M^n#Σ^n does not always results in a new manifold. Following de Sapio <cit.>  (recall from <cit.> and <cit.> that, in de Sapio's notation, Σ^8∈σ(π_3SO(4),π_4SO(3)), Σ^10 is an order 3 generator of σ(π3SO(6),π_6SO(3)) and Σ^4m+1=σ(τ_2m,τ_2m), where τ_n:S^n→ SO(n+1) is a transition function for the tangent bundle of S^n+1), one concludes that M^n#Σ^n≅ M^n if M^n is the non-trivial SU(2)-principal bundle over S^5, the SU(2)-principal bundle over S^7 corresponding to a generator of π_6SU(2)≅ Z_12 and the unit sphere bundle T_1S^2m+1. M^n#Σ^n M^n for the following M^n: any product of standard spheres; any 4-sphere bundle over S^4; Σ^8× S^2; the SU(2)-principal bundles corresponding to three times a generator in π_6SU(2). All other cases are undecidable within the authors reach.
	
	
	
	

 §.§ Cross Diagrams and Notation
 
	The present construction encompasses the constructions in <cit.>. The aim is to produce a diagram as in (<ref>) out of a given  G-manifold M and equivariant bundle data (see Definition <ref>  for details). For short, we denote diagram (<ref>) by Mπ←Pπ'→M'.
	
	In diagram (<ref>), P is a (special) G-G-bundle: following <cit.>, a G-G-bundle π:P→ M is a principal G-bundle equipped with an extra G-action, which we call the ⋆-action. Both G-actions on P are assumed to commute, making P a G×G-manifold. When dealing with the G× G-action, we use G×{𝕀} as the ⋆-action and {𝕀}× G as the principal action of π. We denote the ⋆-action (respectively, the principal action of π) by left juxtaposition (respectively, right juxtaposition). That is, for every (r,s,p)∈ G× G× P, (r,s,p)↦ rps^-1 (we use the inverse of s to keep the `left action' convention).
	
	Since the ⋆-action commutes with the principal action of π, it descends to a G-action on M which we denote by (r,x)↦ r· x. The action makes π equivariant: π(rps^-1)=rπ(p). In particular, if (r,s)∈ G× G fix p, r fixes π(p). On the other hand, if r∈ G fixes x∈ M, since the principal action of π acts freely and transitively on the fibers of π, for every p∈π^-1(x), there is a unique s_p∈ G such that rps_p^-1=p. Note that s_pg^-1=gs_pg^-1. We denote by (G× G)_p and G_x the isotropy subgroups on P and M.
	
	
	
	As in <cit.>, we consider only special G-G-bundles (and, by doing so, omit the adjective `special') by imposing two further conditions: the ⋆-action must be free; for every p∈ P, there is g∈ G such that[In practice we use Definition <ref>, based on transition functions. An equivalence between (<ref>)  and Definition <ref> follows from Proposition <ref>.]
	
    (G× G)_p={(h,ghg^-1) | h∈ G_π(p)}.

	Observe that, if Mπ←Pπ'→M' is a special G-G-bundle, so it is M'π'←Pπ→M (by interchanging the wholes of the two actions). In particular, the principal action of π defines a G-action on M'.
	
	
	Most manifolds will be constructed by patching together subsets. For A_i⊂ X_i, i∈Λ, and  bijections f_ij:A_i→ A_j satisfying f_ikf_kjf_ji(x)=f_ii(x)=x (for any x it make sense), we denote
	{f_ij:A_i→ A_j}_i,j∈Λ, we denote 
	
    ∪_f_ijX_i=∪X_i/∼,
 
	where ∼ is the equivalence relation whose non-trivial relations are A_i∋ x∼ f_ij(x)∈ A_j. Of particular interest are the cases of `twisted manifolds' and of principal bundles: let {X_i=U_i}_i∈Λ be an open cover for a manifold M and f_ij:U_i∩ U_j→ U_i∩ U_j diffeomorphisms. The new object ∪_f_ijU_i is a manifold locally resembling M, but with possibly different global nature; to define a principal bundle, it is sufficient to provide a collection of transition functions. That is,  a collection {ϕ_ij:U_i∩ U_j→ G}_i,j∈Λ satisfying the cocycle condition:
	
    ϕ_ijϕ_jkϕ_ki(x) =ϕ_ii(x)=id∈ G,  ∀ x ∈ U_i∩ U_j ∩ U_k.

	The total space P=∪_f_ijX_i is recovered by setting X_i=U_i× G and f_ij(x,g)=f_ϕ_ij(x,g)=(x,gϕ_ij(x)).  The projection π : P→ M and the principal action (s,p)↦ ps^-1 are defined on U_i× G by
	
    π(x,g)=x,     (x,g)s^-1=(x,sg).

	Both π and the multiplication (as defined above) commute with f_ϕ_ij, defining global maps.
	
	Although, only open covers are considered above, the case  where M is the union of two manifolds by their boundaries is present in the examples. Let M=X∪ Y and f:∂ X→∂ Y a diffeomorphism. The manifold M'=X∪_f Y admits a unique smooth structure such that the inclusions X,Y⊂ M' are smooth. Such smooth structure is obtained through a collaring argument:   using the collaring theorem (Theorem (3.3) in Kosinski <cit.>) one can extend X,Y to open manifolds X',Y' (without boundaries) whose 'ends' are diffeomorphic to ∂ X× (-1,1), ∂ Y× (-1,1) with embeddings j_X:∂ X× (-1,1)→ X', j_Y:∂ Y× (-1,1)→ Y'. Then M' is diffeomorphic to X'∪_f' Y' where f'(j_X(x,t))=j_Y(f(x̃),-t). Isotopies of f defines isotopies of f' implying that the manifold structure of M' does not depend on the choices of j_X,j_Y or the isotopic class of f.
	
	
	
	
	A sphere bundle S^k-1⋯ M→ B is called linear if O(k) (acting in the usual way on S^k-1) is a structure group. Equivalently, if there is a set of transition functions {ϕ_ij:U_i∩ U_j→ O(n)}.
	
	Recall that linear sphere bundles can always be suspended: consider s_k:O(k)→ O(k+1)  the inclusion of O(k) as the subgroup of O(k+1) with 1 in the upper-left corner. If {ϕ_ij:U_i∩ U_j→ O(k)} are transition functions for S^k-1⋯ Mπ→B, then {s_kϕ_ij:U_i∩ U_j→ O(k+1)} are transition functions for a linear S^k-bundle over B.
	
	We denote the quaternionic field as H and its subspace of pure imaginary quaternions as Im H. The norm, inverse and conjugate of a quaternion x are denoted as |x|, x^-1, x̅, respectively.
	
	Some standard spaces (such as discs and spheres) are described in coordinates. When we denote M⊂ V_1× V_2× V_3 we mean that a point in M is a triple (x_1,x_2,x_3) where x_i∈ V_i – we observe there might be relations among the x_i's. For instance, S^7⊂ H× H denotes S^7={(x,y)∈ H× H |  |x|^2+|y|^2=1}.
	
	We denote by R_g the Riemannian tensor of the metric g, adopting the sign convention 
	
    R_g(X,Y)Z=∇_X∇_YZ-∇_Y∇_XZ-∇_[X,Y]Z.

	We denote K_g(X,Y)=g(R_g(X,Y)Y,X), the unreduced sectional curvature of g and by ||X∧ Y||^2_g=||X||^2_g||Y||^2-g(X,Y)^2. The Ricci tensor of g is denoted by
	
    _g(X,Y)=∑_i=1^ng(R(e_i,X)Y,e_i),

	where {e_1,...,e_n} is an orthonormal basis for g. The associated quadratic form is denoted _g(X)=_g(X,X).
	
	
	
	
	
	
	
	
	
	
	
	

 §.§ Acknowledgments
 Part of this work was accomplished during the first author's PhD under Prof. A. Rigas and Prof. C. Durán and a postdoc period under Prof. L. A. B. San Martin. The first author thanks all of them.
	
	Both thank Prof. Lino Grama  for suggestions and comments. The second author also thanks Prof. L. Grama for teaching significant part of his geometrical background.
	
	

§ CONSTRUCTING G-G-BUNDLES

	
	
	
	Consider a G-manifold M, an open cover {U_i} and a set of transition functions {ϕ_ij:U_i∩ U_j→ G}. By imposing conditions on U_i and ϕ_ij, one can end up with extra structure on the bundle π:P→ M defined by {ϕ_ij}.
	
	
		Let M be a G-manifold and {U_i} a G-invariant open cover (i.e., U_i is G-invariant for every i∈Λ). A collection {ϕ_ij:U_i∩ U_j→ G} is called a ⋆-collection (or, for short, a ⋆-cocycle) if ϕ_ij satisfies the cocycle condition (<ref>) and 
		
    ϕ_ij(g· x) = gϕ_ij(x)g^-1,

		for all i,j and x∈ U_i∩ U_j.
	
	
	Condition (<ref>) is just equivariance with respect to conjugation on G. Given ϕ_ij, we define the adjoint map 
	
    ϕ_ij : U_i ∩ U_j   → U_i ∩ U_j
    
    	x   ↦ϕ_ij(x)· x.

	ϕ_ij is a G-equivariant diffeomorphism of U_i∩ U_j whenever ϕ_ij satisfies (<ref>) (Lemma <ref>). 
	
	The core of the paper resides on the next Theorem.
	
	
	
		Let π:P→ M be the principal G-bundle associated to a ⋆-collection {ϕ_ij:U_i∩ U_j→ G}. Then P admits a new action ⋆:G× P→ P, such that
		
			
  * ⋆ is free and commutes with the principal action of π
			
  * the quotient P/⋆ is a G-manifold  G-equivariantly diffeomorphic to
			
    M':= ∪_ϕ_ijU_i,

			where the G-action on U_i is the restriction of the G-action on M.
		
	
	
	
		Let P=∪_f_ϕ_ij U_i× G be as in section <ref>. We define the ⋆-action as
		
    r(x,g)=(r· x,gr^-1).

		The action is globally well-defined since, for  x∈ U_i∩ U_j,
		
    (q· x,gq^-1ϕ_ij(qx)) = (q· x,gϕ_ij(x)q^-1).

		The definition of M' implicitly requires the cocycle condition  ϕ_jkϕ_ij=ϕ_ik. Moreover, its smooth G-manifold structure requires  ϕ_ij to be equivariant diffeomorphisms. Next Lemma guarantees both conditions (see also Bredon <cit.>).
		
			Let U be a smooth G-manifold and θ, θ': U → G be smooth maps satisfying (<ref>). Then θ,θ':U→ U are G-equivariant diffeomorphisms and θθ'=θ'θ, where θθ': U → G is the pointwise multiplication θθ'(x) = θ(x)θ'(x). 
			
			
		
		
			For x ∈ U,
			
    (θ'θ)(x) = θ'(θ(x)x)θ(x)x= θ(x)θ'(x)θ(x)^-1θ(x)x=θ(x)θ'(x)x=θθ'(x).

			In particular, taking θ'(x)=θ(x)^-1, one gets θ'=θ^-1. G-equivariance of θ is straightforward.
		
		To identify P/⋆ with M', we define  π':P→ M' in each  U_i× G by
		
    π'(x,g)=g· x.

		(compare Cheeger <cit.>) π' is well defined on P since, for all x∈ U_i∩ U_j, 
		
    π'(x,gϕ_ij(x))=gϕ_ij(x)x=ϕ_ij(gx)gx=ϕ_ij(π'(x,g)),

		Moreover, π'(rp)=π'(p) for every p∈ P and ι:U_i→ P defined by 
		
    ι_i(x)=(x,𝕀)
 defines a local section for both π and π'. In particular, π' is a submersion whose fibers are the ⋆-orbits.	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	

 §.§ Induced Bundles

	
	
	Before proceeding to examples, we provide a method to produce new G-G-bundles out of old ones. As one can see, a G-G-bundle naturally lies on the category of G-spaces and G-equivariant maps. Therefore, one could expect that the pullback construction could be carried out by equivariant maps on  the set of G-G-bundles. Here we provide details of this procedure (compare <cit.>).
	
	Let M, N be smooth G-manifolds and  f : N → M a smooth  G-equivariant map, i.e., f(g· x) = g· f(x) for every (x,g) ∈ N× G. Let {ϕ_ij:U_i∩ U_j→ G} be a ⋆-collection of transition functions  associated to the covering M=∪ U_i and consider π : P → M the ⋆-bundle associated to such collection. We define the induced bundle (or pullback bundle) π_f : f^∗P → N  by
	
    f^∗P := {(x,p) ∈ N× P : f(x) = π(p)},

	with projection π_f(x,p) := x and principal action
	
    (x,p)s^-1 := (x,sp).

	Standard arguments shows that f^*P is a submanifold of N× P and π_f is a smooth principal submersion. The ⋆-action can also be pulled back to π_f. Define
	
    r(x,p) := (r· x,gr^-1).

	Equation (<ref>) produces an action on N× P that leaves f^∗P invariant. π_f is clearly equivariant with respect to (<ref>).
	
	The induced bundle construction becomes a more interesting construction due to its relation with the original bundle π:P→ N.
	
	 Let f:N→ M be a smooth G-equivariant map and Pπ→ M the G-G-bundle associated to {ϕ_ij:U_i∩ U_j→ G}. Then,
		
			
  * π_f:f^*P→ N is equivariantly isomorphic to  the G-G-bundle associated to the ⋆-collection {ϕ_ij∘ f : f^-1(U_i) ∩ f^-1(U_j) → G};
			
  * the quotient f^*P/⋆ is G-equivariantly diffeomorphic to  
			
    N' := ∪_ϕ_ij∘ ff^-1(U_i);

			
  * there is a well-defined map f':N'→ M' such that, f'|_f^-1(U_i)=f|_f^-1(U_i).
		
	
	
		The G-equivariance of f guarantees the invariance of the sets f^-1(U_i) and equivariance of ϕ_ij∘ f:
		
    ϕ_ij(f(g· x))) = ϕ_ij(g· f(x)) = gϕ_ij(f(x))g^-1.

		To show that f^*P is the bundle associated to {ϕ_ij∘ f}, consider P=∪_f_ϕ_ij U_i× G and define a map Φ_i:π_f^-1(f^-1(U_i))→ f^-1(U_i)× G via the expression
		
    (x,(f(x),g))↦ (x,g).

		The maps Φ_i  clearly patch together to define a G×G-equivariant diffeomorphism between f^*P and ∪_f_ϕ_ijf (f^-1(U_i)×G). 
		
		Item (ii)  follows from  item (i) and the second item in Theorem <ref>.  For item (iii), consider f^*:f^*P→ P, f^*(x,p)=p. f^* is the map the makes the following pullback diagram commutative:
		
    f^*P[r]^f^*[d]^π_f    P[d]^π
     N[r]^f     M

		One observes that f^* is G× G-equivariant. In particular, it defines a map f':f^*P/⋆→ P/⋆. It follows from (<ref>) that f^*(ι_i(x))=ι_i(f(x)), where ι_i is defined in (<ref>). Therefore, for x∈ f^-1(U_i),
		
    f'(x)=π(f^*(ι_i(x)))=π(ι_i(f(x)))=f(x).
 
	
	
	
	

§ EXAMPLES

	
	
	
	
	
	
	In this section we provide basic examples of (special) G-G-bundles. 
	We start with the Hopf fibration h:S^7→ S^4 and linear S^3-bundles over S^4, followed by the bundles in <cit.> that realizes  homotopic 8-, 10- and Kervaire spheres (see also <cit.>). 
	
	Examples <ref> and <ref> can be found in <cit.>. The bundles in Wilhelm <cit.> also can be described as G-G-bundles.
	
	

 §.§ The Hopf S^3-S^3-bundle

	Consider S^7,S^4 as  the unitary spheres on the quaternionic plane ℍ× H and on R× H, respectively. Define the Hopf map h:S^7→ S^4 by
	
    h[ x; y ]=[ |x|^2-|y|^2;        2xy̅ ].

	Let S^3 denote the unitary sphere on H and observe that, for r,s∈ S^3,
	
    h[ rxs̅; rys̅ ]=[ |x|^2-|y|^2;     r2xy̅r̅ ].

	Condition (<ref>) is easily verified (notice that it is sufficient to consider x∈ R). Therefore h defines an S^3-S^3-bundle with ⋆-action defined by the r-multiplication. Since quaternionic conjugation on S^7 interchanges r and s, the quotient S^7/⋆ is diffeomorphic to S^4.
	
	
	
	Consider D^4_±={(λ,x)∈ S^4 | ±λ≥ 0}.  Local trivializations of h are given by the maps Φ_±:D^4_±→ S^7,
	
    Φ_+[ [   cos (t); sin (t) ξ ], q ]=[    cos (t/2)q̅; sin (t/2) ξ̅q̅ ],  Φ_-[ [  -cos (t); sin (t) ξ ],  q ]=[ sin (t/2)ξq̅; cos (t/2) q̅ ],

	where ξ is an unitary quaternion. Taking U_0=D^4_+ and U_1=D^4_-, we have ϕ_01:U_0∩ U_1={(0,ξ)∈ S^4 |ξ∈ H}→ S^3 as ϕ_01(0,ξ)=ξ. By identifying U_0∩ U_1=S^3 (dropping the first coordinate), we denote ϕ_01=I:S^3→ S^3, the identity map.
	
	
	

 §.§ The Gromoll-Meyer sphere

	We recall the definition of the Lie group of quaternionic matrices Sp(2)
	
    Sp(2) = {[ a c; b d ]∈ S^7× S^7 |  c̅a + d̅b = 0}.
 
	The projection onto the first column pr:Sp(2)→ S^7 is a principal S^3-bundle with principal action:
	
    [ a c; b d ]q̅ = [  a cq;  b dq ].

	Gromoll and Meyer <cit.> introduced the ⋆-action 
	
    q [ a c; b d ] = [ qaq  qc; qbq  qd ].

	whose quotient is an exotic 7-sphere, concluding their celebrated result on the existence of an exotic sphere with  non-negative sectional curvature:
	The corresponding action on S^7 can be ready from the first column of (<ref>):
	
    q·[ x; y ]=[ qxq̅; qyq̅ ]

	The S^3-S^3-bundle defined by (<ref>),(<ref>) gives rise to the cross-diagram in Durán <cit.> which is used to geometrically produce an explicit clutching diffeomorphism b̂:S^6→ S^6 for Σ^7=Sp(2)/⋆:
	
    S^3@.[d]^∙   
     S^3@..[r]^⋆   Sp(2)[d]^pr[r]^pr'   Σ^7
       S^7

	In order to produce b̂, the geometry of Σ^7 is explored through (<ref>). This paper is dedicated to further advance the geometrical/topological relations started in <cit.>.
	
	As an S^3-S^3-bundle, pr can be realized as a pullback from h:S^7→ S^4. In <cit.>, this pullback realization is used to recover the identification of Sp(2)/⋆ with the Milnor bundle M_1,-2≅ M_2,-1 (as in Gromoll–Meyer <cit.>) which is  an exotic sphere.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	

 §.§ Milnor bundles

	The usual boundary map in the long homotopy sequence of the fibration EG→ BG, G=SO(k) provides a bijection between the set o linear S^k-1-bundles over S^n and π_n-1SO(k). The linear S^3-bundles over S^4 are usually called Milnor bundles. As in Milnor <cit.>, observe that t_mn:S^3→ SO(4),
	
    t_mn(x)v=x^mvx^n,    v∈ H

	are representatives of π_3SO(4)≅ Z⊕ Z. We define M_m,n=D^4× S^3∪_f_t_m,nD^4× S^3. Milnor observed that M_m,n is homeomorphic to S^7 whenever m+n=1, but not diffeomorphic when m=2, for example (see <cit.> for a complete classification). On the other hand, the bundles P_n=M_0,n≅ M_-n,0 are S^3-principal. We use this section to show that every Milnor bundle can be obtained out of some P_n. 
	
	
	
	
	Consider a pair of S^3-principal bundles π_k:P_k→ S^4, π_r:P_r→ S^4. 
	
	π_k is a S^3-S^3-bundle with the  ⋆-action
	
    r(x,q)=(rxr̅,qr̅).

	We consider P_r as the S^3-manifold with action
	
    r· (x,q)=(rxr̅,rqr̅).

	Observe that both (<ref>) and (<ref>) commutes with f_t_n,0 for every n, defining global actions on P_n. Given π_k, its unique trivialization function is ϕ_01(x)=x^k, where both U_0 and U_1 are identified with D^4, the unit disc on H. Consider the S^3-S^3-bundle given by π_r^*P_k→ P_r. Each copy D^4× S^3⊂ P_r is invariant with respect to  (<ref>). The only transition function of π^*_rP_k associated to the cover {D^4× S^3, D^4× S^3} is t_k,0∘π_r|_S^3× S^3:S^3× S^3→ S^3.. 
	
	From Theorem <ref>, the resulting manifold is M'=D^4× S^3∪_ψ D^4× S^3 where ψ is the composition of f_0,r with (t_k,0π_r|_S^3× S^3). A straightforward computation gives ψ=f_t_r,k-r. We conclude that the bundle M_m,n can be realized by a S^3-S^3-bundle P_r← P→ M_m,n, where r=m.
	One sees that r parametrizes the Euler class and k the third homology of M_r,k-r (see Milnor-Stasheff <cit.>). It is worth noticing that (S^4)'=S^4 in S^4← P_k→ (S^4)' and that the map (π_r)':M_r,k-r→ S^4 coincides with the bundle projection π_(r,k-r).The full diagram is:
	
	
	
    @R=9pt@C=7pt@1       S^3 @..[dd]        S^3 @..[dd]   
       S^3@..[rd]        S^3@..[rd]      
          π_r^*P_k[rd][rr][dd]        P_k[rd][dd]   
             M_r,k-r@-[r]     [r]    S^4
           P_r[rr]^π_r       S^4

	The Gromoll–Meyer sphere happens with r=1, k=-1 (in this case, P_1=P_-1=S^7 and π_1=-π_-1=h – it is well known that the pull back of h by -h has total space Sp(2), see <cit.>). 
	
	
	
	
	
		Another way to define a S^3-S^3-bundle over P_r is to consider P_r as the S^3-manifold with action (<ref>). A straightforward computation shows that the resulting diagram is P_r←π_r^*P_k→ P_r-k. 
	
	
	

 §.§ Other exotic spheres

	
	In <cit.>, G-G-bundles were used to give geometric presentations of exotic spheres:
	
	
		There are explicit (special) G-G-bundles:
		
			
  * S^8 π^11⟵E^11→Σ^8, where G=S^3 and Σ^8 is the only exotic 8-sphere
			
  * S^10 π^13⟵E^13→Σ^10, where G=S^3 and Σ^10 is a generator of the order 3 group of homotopy 10-spheres which bound spin manifolds.
			
  * S^2n-1 l_n⟵L_n→Σ^2n-1, where G=O(n) and Σ^2n-1=∂ P^2n(𝐀^2)
		
	
	
	
	
	
	π^11 and π^13 are pulled back from pr:Sp(2)→ S^7 and l_n' from the frame bundle pr_n:O(n+1)→ S^n. As in the case of pr, we consider O(n+1) as a matrix group and pr_n as the projection to the matrix's first column. We give a brief description of  π^11,π^13,l_n.
	
	Consider S^8 ⊂ℝ×ℍ× H endowed with the action 
	
    q·[ λ; x; y ] = [   λ;  qx; qyq ].

	Observe that f_8 : S^8 → S^7, defined   by
	
    f_8[ λ; x; w ] = 1√(λ^2 + |x|^4 + |w|^2)[ λ + xix;       w ],

	is equivariant with respect to (<ref>) and (<ref>). The bundle π^11 is defined as the  pullback of pr:Sp(2)→ S^7 by f_8. 
	
	We present π^10 with two different actions. Let S^10⊂Im H× H× H be the S^3-manifold with one of the following actions[Although action (II) is not considered in <cit.>, going through the proof of Theorem 1 in <cit.>, one verifies that the sphere we get using action (II) is diffeomorphic to Σ^10#Σ^10#Σ^10#Σ^10, which turns to be diffeomorphic to Σ^10.]
	
    (I)  	q·[ p; w; x ] = [   p;  qw; qxq ],       (II)   q·[ p; w; x ] = [ qpq̅; qwq̅; qxq̅ ].

	For the pulling back map, f_10:S^10→ S^7, define the Blakers-Massey element b: S^6 → S^3 as in <cit.>:
	
    b(p,w) := w/|w|exp(π p)w/|w|,  if w ≠ 0,
    
    	-1,  if w = 0.

	Let φ:[0,1]→[0,1] be a smooth non-decreasing function that is the identity on [1/4,3/4] and constant near 0 and 1, fixing 0 and 1. Set	
	
    f_10[ ξ; w; x ]=[ √(1-φ(|x|)^2)b(ξ/√(|ξ|^2+|w|^2),w/√(|ξ|^2+|w|^2));                                       φ(|x|)x/|x| ].

	Note that f_10 is equivariant with respect to (<ref>) and (<ref>). Define π^13 is the pullback of  pr:Sp(2)→ S^7  by f_10.
	
		Action (<ref>)-(I), can be extended to the effective SO(4)=S^3× S^3/{±(1,1)}-action
		
    (q,r)·[ p; w; x ] = [ rpr̅; qwr̅;  qxq ]

		which can be used to realize Σ^10=(S^10)' as a SO(4)-manifold. To this aim, one can either  observe that f_10 is invariant with respect to the r-coordinate (therefore the clutching diffeomorphism will have the desired equivariance) or consider Sp(2)× S^3/{±(1,1)} as an SO(4)-SO(4)-bundle with the r-coordinate acting only on the S^3-factor. According to a Conjecture 2 in Straume <cit.>, the SO(4)-manifold Σ^10 should be a peculiarly highly symmetric sphere among homotopy spheres that not bound parallelizable manifolds (see <cit.>).
	
	
	The O(n)-O(n)-bundle π_n is realized as the pullback of pr_n:O(n+1)→ S^n by Jτ_n (defined below). Observe that pr_n is an O(n)-O(n)-bundle with ⋆-action given by left multiplication of matrices: consider O(n) acting through the morphism s_n:O(n)→ O(n+1)  and define pr_n(A)=Ae_0, where e_0=(1,0,0,...,0)^T. The O(n)-action 
	
    r⋆ A=s_n(r)A

	is free and satisfies pr_n(rA)=s_n(r) pr_n(A). An equivariant transition function for pr_n is τ_n:S^n-1→ O(n) defined as
	
    τ_n(x)v=2x,v-v.
 
	We consider S^2n-1⊂ R^n⊕ R^n and the map Jτ_n:S^2n-1→ S^n, 
	
    Jτ_n(x_1,x_2)= exp_e_0(πτ_n(x_2/|x_2|)x_1).

	Jτ_n is equivariant with respect to (<ref>) and the biaxial action
	
    r·[ x; y ]=[ rx; ry ].

	
	The resulting O(n)-O(n)-bundle l_n: L_n=Jτ_n^*O(n+1)→ S^2n-1 gives rise to the manifold Σ^2n-1=L_n/⋆. 
	Following <cit.>, Σ^2n-1 is homeomorphic to S^2n-1 for n odd and has H_n-1(Σ^2n-1)≅ Z_3 when  n is even (as mentioned in the introduction,  Σ^2n-1=∂ P^2n(A_2) in Bredon <cit.>). Next we highlight two phenomena depending on the parity of n.
	
	If n=2m, O(2m) admits a subgroup S^1⊂ O(2m) acting freely on S^2n-1: consider S^1 as the subgroup of block diagonal matrices diag(A,A,...,A), where A∈ SO(2) (i.e.,  the exponential of the standard  complex structure of R^2m). From Theorem <ref>, item (ii), the resulting  S^1-action is free on Σ^2n-1. The resulting quotient Σ^4m-1/S^1, denoted by Σ P C^2m-1, has π_k Σ P C^2m-1≅π_k CP^2m-1 for k≠ 2m and π_2mΣ P C^2m-1≅ Z_3. 
	
	When n=4m, one gets a free S^3-action by considering A∈ Sp(1)⊂ O(4) in diag(A,...,A). The resulting quotient ,Σ^8m-1/S^3:=Σ P H^2m-1, satisfies π_k Σ P H^2m-1≅π_k HP^2m-1 for k≠ 4m and π_4mΣ P H^2m-1≅ Z_3. We also observe that there are subgroups U(m)⊂ O(2m), Sp(m)⊂ O(4m) commuting with the S^1-, respectively S^3-action, defining Σ P C^2m-1 as a U(m)-manifold and Σ P H^2m-1 as a Sp(m)-manifold. In section <ref>, we provide invariant metrics of positive Ricci curvature on both Σ P C^2m-1 and  Σ P H^2m-1.
	
	If n is odd, n=2m+1, we  consider the bundle reduction pr_m^ C:U(m+1)→ S^2m+1. Considering U(m+1)⊂ O(n+1), we take pr_m^ C as the restriction pr_n|_U(m+1). By observing that both right and left multiplication by U(m)⊂ O(n) leaves U(m+1) invariant, one concludes that pr_m^ C is a U(m)-U(m)-bundle. An equivariant transition map τ_m^ C:m:S^2m→ U(m) is presented in <cit.>:
	
	
    τ_m^ C[ y; z ]= (𝕀 -z/|z|(1+e^π y)z̅^t/|z|),
	
	where y∈ i R and z∈ C^m. 	We can therefore consider A U(m)-reduction of L_n can be realized by the pull back of Jτ^ C_m:S^4m+1→ S^2m+1,
	
    Jτ^ C_m(x_1,x_2)= exp(πτ^ C_m(x_2/|x_2|)x_1).

	Jη_m is equivariant with respect to (<ref>) and the U(m)-action defined by restricting  (<ref>) to U(m)⊂ O(n-1). 
	
	If n=4m+3, the analogous reduction Sp(m+1)⊂ O(n+1) works along the same lines, with transition map τ^ H_m:S^4m+2→ Sp(m), obtained by replacing i R by Im H and C^m by H^m in (<ref>). 
	
	The advantage of the U(m), Sp(m) realization of Σ^4m+1 is the presence of fixed points, allowing us to perform equivariant connected sums (see section <ref>).
	
	

§ CONNECTED SUMS

	Here we realize  diagrams of the form M← f^*P→ M#Σ^n, where Σ^n is an exotic sphere. In  section <ref>, the diagram is used to study the Ricci curvature of M#Σ^n  avoiding the intricate geometry of a connected sum.
	
	
	
	Let M be a G-manifold with a fixed point p∈ M. Assume  G is compact and that M has a G-invariant Riemannian metric. The differential of the action at p induces a morphism ρ:G→ O(T_pM) called the isotropic representation of G at p. 
	
	On the other hand, we consider S^n⊂ R× T_pM as a G-manifold with action g· (λ,x)=(λ,ρ(g)x). Note that  e_0=(1,0)^T is a fixed point. Consider D_±={(λ, x)∈ S^n |±λ≥ 0} and S^n-1=D_+∩ D_-, observing that D_+,D_-,S^n-1 are G-invariant subsets. If ϕ:S^n-1→ G satisfies the equivariant  condition (<ref>), one constructs the G-G-bundle P→ S^n 
	
    P=D_+× G⋃_f_ϕ D_-× G,

	where f_ϕ(x,g)=(x,gϕ(x)) and 
	
	the ⋆-action is defined as in Theorem <ref>: r(x,g)=(r· x,gr^-1).
	Using (<ref>), Theorem <ref> identifies  (S^n)' as the twisted sphere   (S^n)'=D^n∪_ϕ̂ D^n. 
	
	As a next step, we pull back P to M. The pullback function f:M→ S^n can be obtained along the lines of the Thom–Pontrjagyn construction (see Kosinski <cit.>, sections IX.4 and IX.5): let D_ϵ be an ϵ-disc around the fixed point p such that exp_p|_D_ϵ is a diffeomorphism. Define f:M→ S^n as
	
    f(x)=exp_e_0(πφ(|exp_p^-1(x)|/ϵ)exp_p^-1(x)/|exp_p^-1(x)|),     x∈ D_ϵ
     -e_0,     x∉ D_ϵ .
 
	
	As P is trivial along D_±, Proposition <ref>, item (i), gives
	
    f^*P=f^-1(D_+)× G⋃_f_ϕ∘ f f^-1(D_-)× G=(M-D_ϵ/2)× G⋃_f_ϕ D_ϵ/2× G

	where D_ϵ/2 is the ϵ/2-disc around p and we identify S^n-1 with ∂ D_ϵ/2. Therefore,
	
    M'=(M-D_ϵ/2)⋃_ϕ̂ D_ϵ/2.

	M' is easily seem to be diffeomorphic to M# (S^n)'. We have:
	
	
		Let M^n be  G-manifolds and p∈ M  with isotropic representation ρ:G→ O(n). Then, given a map ϕ:S^n-1→ G satisfying 
		
    ϕ(ρ(g)x)=gϕ(x)g^-1,

		there is a G-G-bundle M← P→ M#Σ^n, where Σ^n=D^n∪_ϕ̂D^n.
	
	
	Following Lemma <ref>, of ϕ satisfies the hypothesis on Theorem <ref>, so does ϕ^k, for every k∈ Z. Since #_kΣ^n=D^n∪_ϕ^kD^n, the resulting M' gives the k-fold connected sum M#Σ^n#...#Σ^n.
		
	
		One can use the Thom–Pontrjagyn to provide a more general construction: suppose N^k⊂ M^n+k is a G-invariant manifold and F: R^n× N→ M is a frame of ν N such that G satisfies gF(v,x)=F(ρ(g)v,gx) for some linear representation ρ:G→ O(n). One considers the  f:M→ S^n+1,
		
    f(z)=exp_e_0(πφ(|v|)v/|v|),     z=F(x,v)
     -e_0,     z∉ F( R^k× N) .
 
		Given a function ϕ: S^n-1→ G as in Theorem <ref>, one can again consider the G-G-bundle (<ref>) and its pull-back f^*P. The resulting manifolds, M', resembles Bredon's pairing <cit.> as a version of M `twisted' along the boundary of a tube around N. Examples based on this construction will be presented elsewhere.
		
	
	
	
	It follows  that  there are functions ϕ:S^n-1→ G realizing  P=D^n× G∪_f_ϕ D^n× G and, therefore, Σ^n=D^n∪ _ϕD^n, where  P is E^11, E^13 and L_2m+1 in Theorem <ref> (Proposition <ref> or Theorem 4.2 and Proposition 4.4 in <cit.>). Explicit formulas for ϕ in the case of E^11 and E^13 are presented in <cit.>. For b:S^6→ S^3 in <ref>, Σ^7=D^7∪_b̂ D^7 is a generator of the group of homotopy 7-spheres (<cit.>). 
	
	Observe that the representations (ρ_n) in the introduction are related to the fixed points of the S^3-manifolds Σ^7,Σ^8,Σ^10, the U(m)-manifold Σ^4m+1 and the Sp(m)-manifold Σ^8m+5, respectively. To realize the examples in Theorem <ref> we just need to observe that M^n  in Theorem <ref> has a G-action carrying a fixed point with isotropy representation (ρ_n).
	
	
	
	
	
	
	
	
	
	
	
	The reducibility of the representations (ρ_7)-(ρ_8m+5) allows us to explore G-manifolds which are locally products, such as (trivial and non-trivial) bundles. In what follows, we list some examples. As `building blocks', we use  homogeneous manifolds such as S^n  and projective spaces CP^m, HP^m. If H/K is an homogeneous manifold, the pair (H/K,ρ') is to be interpreted as the manifold H/K endowed with the G-action  induced by ρ'(G)⊂ K. Thus, ρ' is the isotropy representation of the G-manifold H/K at the `base point'  K∈ H/K. In general, (N,ρ') will denote a manifold that posses a fixed-point with isotropic representation ρ'. For the representations, we use the notation in the list (ρ_7)-(ρ_8m+5).
	
	

  §.§.§ Product manifolds

	
		
  * (S^3,ρ_1)× (S^4,ρ_1⊕ρ_0)
		
  * (S^3,ρ_1)× (S^3,ρ_1)× (N^1,ρ_0)
		
  * (S^6,ρ_1⊕ρ_1)× (N^1,ρ_0)
		
  * (S^3,ρ_1)× (S^5,ρ_⊕ρ_0)
		
  * (S^3,ρ_1)× (S^4,ρ_)× (N^1, ρ_0)
		
  * (S^7,ρ_⊕ρ_1)× (N^1,ρ_0)
		
  * (N^8,ρ_⊕ρ_1⊕ρ_0)× (N^2,2ρ_0)
		
  * (S^2m,ρ_U(m))× (S^2m+1,ρ_U(m)⊕ρ_0)
		
  * (S^4m,ρ_U(m)⊕ρ_U(m))× (N^1,ρ_0)
		
  * ( CP^m,ρ_U(m))× (S^2m+1,ρ_U(m)⊕ρ_0) 
		
  * ( CP^m,ρ_U(m))× (S^2m,ρ_U(m))× (N^1,ρ_0) 
		
  * ( CP^m,ρ_U(m))× ( CP^m,ρ_U(m))× (N^1,ρ_0) 
		
  * (U(m+2)/SU(2)× U(m),ρ_U(m)⊕ρ_U(m)⊕ρ_0)
		
  * (U(m+2)/U(2)× U(m),ρ_U(m)⊕ρ_U(m))× (N^1,ρ_0)
		
  * (S^4m,ρ_Sp(m))× (S^4m+5,ρ_Sp(m)⊕ 5ρ_0)
		
  * (S^8m,ρ_Sp(m)⊕ρ_Sp(m))× (N^5,5ρ_0)
		
  * ( HP^m,ρ_Sp(m))× (S^4m+l,ρ_Sp(m)⊕ lρ_0)× (N^5-l,(5-l)ρ_0), for 0≤ l≤ 5 
		
  * ( HP^m,ρ_Sp(m))× ( HP^m,ρ_Sp(m))× (N^5,5ρ_0) 
		
  * (Sp(m+2)/Sp(2)× Sp(m),ρ_Sp(m)⊕ρ_Sp(m))× (N^5,5ρ_0)
		
	
	
		As mentioned, T_1S^2m+1#Σ^4m+1≅ T_1S^2m+1 (see de Sapio <cit.>).
	
	
	

 §.§ Bundles over spheres

	To consider actions on bundles, we rely on explicit equivariant expressions of transition functions, such as (<ref>).
	
	

  §.§.§ Milnor bundles
 A family of examples are given by the Milnor bundles:  action (<ref>)  is well defined in every M_m,n and  (0,1)^T is a fixed point with isotropy representation (ρ_7).
	
	

  §.§.§ Explicit non-linear S^6-bundles over S^1
 Given a diffeomorphism h:S^6→ S^6, a S^6-bundles over S^1 is defined by
	
    E_h=[0,1]× S^6∪_f_h[0,1]× S^6

	where f_h:{0,1}× S^6→{0,1}× S^6 is defined by f_h(0,x)=(0,x), f_h(1,x)=(1,h(x)). There are interesting choices we can make for h: by taking h=α, the antipodal map in S^6 one gets the non-trivial linear S^6-bundle  over S^1. Moreover, <cit.> provides the family S^3-equivariant  fixed-point free involutions θ_k=αb̂^k:S^6→ S^6. It is also proved in <cit.> that α=θ_0,...θ_27 parametrizes the 28 connected components of Diff^-(S^6), the set of orientation-reversing diffeomorphisms of S^6. 	Therefore, E_k=E_θ_k is isomorphic to a linear bundle if and only k≡0 28. 
	
	One can define E_k as an S^3-manifold using action (<ref>) on S^6 (taking S^6=S^7∩{(x)=0}). It has a fixed point ((0,(0,1)^T) in any copy of [0,1]× S^6) with isotropy representation (ρ_7). The authors don't known whenever E_k is diffeomorphic to E_0 or not.
	
	The advantage of using θ_k instead of the orientation-preserving b̂^k is that θ_k, being a fixed-point free involution, defines a free Z_2-action on E_k. The quotient is the product of a fake projective plane F RP^6 and S^1.
	
	Although the observations above does provide S^3-S^3-bundles E_k← P→ E_k#Σ^7, we are not able to give any new relevant information about E_k#Σ^7 since the geometry of E_k is itself unknown to the authors (for instance, E_k can't have positive Ricci curvature since it has infinite fundamental group). One might believe that E_k≅ E_0#_kΣ^7. In this case, there is a cross-diagram E_0← P→ E_k and, possibly, something could be said about E_k.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	

 §.§ 8-dimensional bundles

	Linear S^3-bundles over S^5 and S^4-bundles over S^4 are parametrized by π_4SO(4)≈ Z_2+ Z_2 and π_3SO(5)≈ Z, respectively. Define:
	
    η:S^4   → S^3 
    [ λ; x ]   ↦λ +xix̅/|λ +xix̅ |

	Then η_L=ρ_L∘η and η_R=ρ_R∘η are generators for π_4SO(4), where ρ_L,ρ_R denotes the 4-dimensional representation of S^3 defined by left, respectively right, multiplication by quaternions. ρ_1⊕ 2ρ_0=I_5:S^3→ SO(5) generates π_3SO(5). Note that  η_L,η_R  are equivariant with respect to ρ_1⊕ 2ρ_0 and I_5(qxq̅)=I_5(q)I_5(x)I_5(q)^-1. We get the bundles
	
    P_η_ϵ=D^5× S^3⋃_f_η_ϵD^5× S^3,     P_I_5^k=D^4× S^4⋃_f_I_5^kD^4× S^4,

	where ϵ=L,R and I_5^k(q)=I_5(q)^k. Both bundles admit the actions 
	
    q·([ λ; x ], g)=([  λ; qx ], qgq̅),     q·([ λ_1; x_1 ],[ λ_2; x_2 ])=([    λ_1; qx_1q̅ ], [  λ_2; qx_2 ]).

	Therefore P_η_ϵ,P_I_5^k have fixed points with isotropy representation (ρ_8).
	
	
		The analogous bundle P_η_Lη_R, whose transition function is given by the product η_Lη_R(z)=η_L(z)η_R(z) and P_I_5^k, k≠ 0 2 are stabilized by Σ^8, i.e., M#Σ^8≅ M, for M=P_η_Lη_R,P_I^k_5 (see de Sapio <cit.> – recall that Σ^8=σ_4,3(η_Lη_R,I_5) and that θ^8≈ Z_2). If k is even, P_I_5^k#Σ^8 P_I_5^k (see de Sapio <cit.>). The authors does not known whenever P_η_ϵ is stabilized or not by Σ^8.
	
	
	
		Another S^3-manifold with istropy  (ρ_8) is HP^2 with an S^3=Sp(1)-action derived from a suitable subgroup of Sp(2)× Sp(1),  its standard isotropic representation. However, the promising manifold HP^2#Σ^8 is diffeomorphic to HP^2 (as pointed out to the first author by D. Crowley – see Kramer–Stolz <cit.> for a reference). 
	
	
	

  §.§.§ 10 dimensional bundles

	The case of S^3-bundles over S^7 is of special interest. We deal with the principal case for simplicity. It is known that b:S^6→ S^3, defined in (<ref>), is a generator of π_6S^3 (it is obtained as an explicit clutching map of pr:Sp(2)→ S^7 in <cit.>). It admits the following SO(4) symmetry:
	
    b[ rpr̅; qwr̅ ]=qb[ p; w ]q̅.

	Therefore, the map f_b:S^6× S^3→ S^6× S^3 is equivariant with respect to the (S^3)^3-action:
	
    (q,r,s)·([ p; w ], g)=([ rpr̅; qwr̅ ], sgq̅),

	defining P_b^k=D^7× S^3∪_f_b^kD^7× S^3 as a (S^3)^3-manifold. P_b^k satisfies (c) by restricting the action to the diagonal {(r,r,r) | r∈ S^3}. A generic linear S^3-bundles is obtained using the transition map L_b^n∘ R_b^m. Denote this bundle by P^10_m,n→ S^7. Since f_L_b^n∘ R_b^m is still equivariant with respect to the diagonal action,  P^10_m,n satisfies (c).
	
	
		Recalling from <cit.> that Σ^10=σ_3,6(I_6,L_b∘ R_b), one concludes that P_b^k#Σ^10≅ P_b^k if and only if k≠ 0 3 (see de Sapio <cit.>). The authors were not able to find out whether P_m,n^10#Σ^10 is diffeomorphic to P_m,n^10 or not. 
	
	
		The diffeomorphism Θ_10 in <cit.> provides an explicit representative of a class in π_3(Diff^+(S^6)) not represented by π_3(O(7)). More specifically, it furnishes a non-linear S^6-bundle over S^4 whose total space is an S^3-manifold with a fixed point with isotropy (ρ_10). Note that the works of Nash <cit.> or Poor <cit.> does not provide positive Ricci on this bundle (supposing its total space is not diffeomorphic to a known space).
	
	
	
	
	
	
	
	

§ THE GEOMETRY OF CROSS-DIAGRAMS

	
	
	Let Mπ←Pπ'→ M' be a G-G-bundle. An efficient way to compare geometries in M and M' is to endow P with a G× G-invariant metric. In this case, the space H”⊂ TP, orthogonal to the G× G-orbits on P, descends isometrically to both H and H', the spaces orthogonal to the G-orbits on M and M', respectively.
	
	In what follows, we provide more details about existence of G× G-invariant metrics on P and explore the transversal geometry induced by the isometries Hdπ← H”dπ'→ H'. We suppose  G compact with a bi-invariant metric originating from an adjoint invariant inner product Q on g. 
	
	There are three different ways to endow P with a G× G-invariant metric:
	
		
  * Since G× G is compact, averaging any initial metric g_0 on P gives an invariant metric g (see Bredon <cit.> for reference)
		
  * A more concrete metric can be obtained our examples: given a map f:N→ M, f^*P is naturally a submanifold of N× P (equation (<ref>)). If M and P are equipped with a G-invariant and a G× G-invariant metric, respectively, the induced metric on f^*P is G× G-invariant. Most of our examples are the pullback of either pr:Sp(2)→ S^7 or the frame bundle pr_n:O(n+1)→ S^n, which admit natural G× G-invariant metrics
		
  * Given a connection 1-form ω:TP→ g and a metric g_M on M, one can endow P with the Kaluza-Klein metric 
		
    ⟨ X, Y⟩ := g_M(dπ X, dπ Y) + Q(ω(X),ω(Y)).
		If both g_M and ω are G-invariant (in the sense that ω_gp(gX)=ω_p(X) for all X∈ TP) then, , is G× G-invariant
	
	
	We focus on (3) and prove:
	
		There exists a connection 1-form ω:TP→ g such that
		
    ω_rp(rX)=ω_p(X)

		for all X∈ TP and r∈ G. Moreover, if g_M is a G-invariant metric, then the metric (<ref>) is G× G-invariant. 
	
	
	It follows immediately:
	
	
		Let M← P→ M' be a G-G-bundle. Then there is a G-invariant metric on M' such that M/G and M'/G are isometric as metric spaces.
	
	
	
	
	First observe there are bijections between the set of orbits of M, M' and P: if x=π(p) and x'=π'(p), then π^-1(Gx)=(G× G)p and π'((G× G)p)=Gx'. The choice of p∈π^-1(x), x∈ Gx and x'∈ Gx' are  irrelevant. Moreover,  if γ: R→ M is a geodesic orthogonal to orbits in M, the identification H← H”→ H' sends γ to a geodesic orthogonal to orbits in M'. We make it explicit in the following proposition (see Proposition 4.4 in <cit.> for the case of a fixed point x).
	
	
		There is an isomorphism Φ:ν Gx→ν Gx' satisfying:  if O⊂ν Gx is such that exp|_ O is a diffeomorphism, then exp|_Φ( O) is a diffeomorphism. 
	
	
	

 §.§ Proof of Proposition <ref>

	We recall that a G-connection on a G-principal bundle is a differential 1-form on P with values on the Lie algebra 𝔤, which is G-equivariant and recognizes the Lie algebra generators of the fundamental vector fields on P. That is, for every ξ∈ g, X∈ TP and q∈ G,
	
		
  * ω(pξ)=ξ
		
  * ω_pg(X g)=_g^-1ω_p(X)
	
	Direct averaging  a connection 1-form by the ⋆-action gives an invariant form: let ω_0 be a connection 1-form for π:P→ M. Given  a Haar measure μ on G with unitary volume, define 
	
    ω_p(X) := ∫_G(ω_0)_gp(gX)dμ.

	Since the ∙- and ⋆-action commutes, it is immediate that ω is a connection 1-form for π and that ω_qp(qX)=ω_q(X) for all X∈ TP, q∈ G.
	

 §.§ Proof of Proposition <ref>

	Given x∈ M, let us describe an isomorphism between ν Gx and ν Gx'. 
	
	
		Given x∈ M, there is p_0∈π^-1(x) such that the isotropy subgroup (G× G)_p_0 is the diagonal Δ G_x={(q,q) | q∈ G_x}.
	
	
		First of all, since π(rps^-1)=rp, if (r,s)∈ (G× G)_p, then r∈ G_x. On the other hand, we can write P=⋃ U_i× G, as the bundle defined  by the ⋆-cocycle {ϕ_ij:U_i∩ U_j→ G }. Suppose x∈ U_i. By definition, the G× G-action on U_i× G is given by r(x,g)s^-1=(rx,sgr^-1). Therefore, for all r∈ G_x, r(x,1)r^-1=(rx,rr^-1)=(x,1).  
	
	Given p_0 as in Claim <ref>,  let Ψ:Gx→ P be defined as Ψ(rx)=rp_0r^-1.  Given X∈ TM and p∈π^-1(x'), denote by L_p(X)∈ H_p the unique horizontal vector such that dπ( L_p(X))=X. The maps Ψ and L_p satisfies:
	
    Ψ(ry)=rΨ(y)r^-1,     L_rps^-1(rX)=r L_p(X)s^-1.

	Define the morphism Φ:ν Gx→ν Gx' as
	
    Φ_rx(X)=dπ'( L_Ψ(rx)(X)).

	
		Φ_rx is an isometry between H and H'.  
	
	
		It is sufficient to show that L_p( H_x)= H”_p, since H”⊂ dπ' and both L_p and dπ'_( dπ')^ are isometries. 
		
		However, since π(rps^-1)=rx, L_p sends vectors tangent to Gx to vectors tangent to (G× G)p. A dimension count shows that L_p( H_x)= H”_p.
	
	Let O⊂ν Gx be such that exp|_ O is a diffeomorphism.  Then Ψ̃:exp( O)× G→ P, defined as
	
    Ψ̃(exp_y(v),g)=exp_Ψ(y)( L_Ψ(y)(v))g^-1,
 
	is a trivialization along exp( O). Moreover,
	
    Ψ̃(exp_ry(rv),sgr^-1)   =exp_Ψ(ry)( L_Ψ(ry)(rv))r(sg)^-1=exp_Ψ(ry)( L_rΨ(y)r^-1(rv))r(sg)^-1
       =exp_Ψ(ry)(r L_Ψ(y)(v)r^-1)r(sg)^-1=rΨ̃(exp_y(v),g) s^-1.

	Therefore, as in (<ref>), π'(Ψ̃(exp_y(v),g))=gexp_y(v), thus, arguments as in the proof of Theorem <ref> guarantees that
	
    exp_y(v)↦π'(Ψ̃(exp_y(v),𝕀))

	defines a diffeomorphism. On the other hand, since π' is a Riemannian submersion and v∈( dπ')^, 
    π'(Ψ̃(exp_rx(v),𝕀))   =π'exp_rxr^-1( L_rxr^-1(v))
       =exp_π '(rxr^-1)(dπ' L_rxr^-1(v))=exp_rx'(Φ(v)).

	
		
		The orbit Gx in Proposition <ref> can be replaced by any G-invariant submnifold N, provided there is a map Ψ:N→ P satisfying Ψ(rx)=rΨ(x)r^-1. The rest of the proof proceeds along the same lines.
		
		
		
		
	
	
		The equivalence of condition (<ref>) and Definition <ref> follows from the equivariance of Φ̃ (equation (<ref>)): let P→ M be a (possibly non-special) G-G-bundle satisfying (<ref>). For every orbit Gx⊂ M, there is an open tubular neighborhood U_x of G_x such that exp: O→ U_x is a diffeomorphism for some O_x⊂ν Gx. Consider the trivialization Φ̃_x:U_x× G→ P as in the proof of Proposition <ref>. The equivariance (<ref>) guarantees that the transition functions related to the open cover {U_x}_x∈ M satisfies the equivariance condition (<ref>).
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	

§ CHEEGER DEFORMATIONS AND RICCI CURVATURE

	
	Lets recall briefly the process known as Cheeger deformation. The main reference are the notes by W. Ziller <cit.> on M. Müter's thesis. The exposition and Theorem <ref> does not intend to be original, but to make the article more self-contained.
	
	Let (M,g) be a Riemannian manifold and (G,Q) a compact Lie group with bi-invariant metric Q acting by isometries on M. For each p ∈ M we can decompose orthogonally the Lie algebra 𝔤 of G as 𝔤 = 𝔤_p⊕𝔪_p, where 𝔤_p denotes the lie algebra of the isotropy group at p. Observe that 𝔪_p is isomorphic to the tangent space to the orbit G· p. 
	
	We call the tangent space T_pG· p as the vertical space at p, 𝒱_p. Its orthogonal complement, H_p   is the horizontal space. Given an element U∈ g, we define its action vector
	
    U^*_p=d/dt|_t=0e^tU· p.

	The map U↦ U^*_p defines linear morphism whose kernel is g_p (the map U↦ U^* defines a Lie algebra morphism between g and the algebra of vector fields X(M) with the vector field brackets). In particular, a tangent vector X at p can be uniquely decomposed  as X = X + U^∗, where X is horizontal and U∈ m_p.
	
	The main idea in the Cheeger deformation is to consider the product manifold M× G, observing that the action 
	
    r(p,g) := (r· p, gr^-1)

	(compare with (<ref>)) is isometric for the product metric g×1/tQ, t > 0. Action (<ref>) is free and its quotient space is diffeomorphic to M (see (<ref>) and (<ref>)). In fact, the projection

    π' : M× G    →⋆\ M× G
    
    	(p,g)    ↦  g· p

	identifies ⋆\ M× G with M, inducing a family of metrics g_t on M. We proceed defining some important tensors:
	
		
  * let P : 𝔪_p →𝔪_p be the orbit tensor of the G-action, defined by
		
    g(U^∗,V^∗) = Q(PU,V),  ∀ U^∗, V^∗∈𝒱_p.
 
		P is a symmetric and positive definite operator 
		
		
  * denote by P_t: m_p→ m_p, the operator 
		
    g_t(U^∗,V^∗) = Q(P_tU,V),   ∀ U^∗, V^∗∈𝒱_p

		
  * we define the metric tensor of g_t,  C_t:T_pM→ T_pM as
		
    g_t(X,Y) = g(C_tX,Y),   ∀X, Y∈ T_pM

	
	The tensors above satisfy:
		
			
  * C_t = 1 in ℋ_p, C_t = P^-1P_t ∈𝒱_p
			
  * P_t = (P^-1 + t1)^-1 = P(1 + tP)^-1
			
  * If X = X + U^∗ then C_t(X) = X + (1 + tP)^-1U^∗
		
	
	
	The advantage of Cheeger deformations is that g_t does not produce `new' planes with zero curvature – in fact, up to a reparametrization of Gr_2(TM) (the Grassmannian of 2-planes of TM), the sectional curvature of a fixed plane is non-decreasing. Let X = X + U^∗, Y = Y + V^∗  tangent vectors. Then κ_t(X,Y) := K_g_t(C_t^-1X,C_t^-1Y) satisfies
	
    κ_t(X,Y) = κ_0(X,Y) +t^3/4[PU,PV]_Q^2 + z_t(X,Y),

	where z_t is a non-negative term related to the fundamental tensors of the foliation (see <cit.>).
	
	
	
	
	
	In what follows, we  study the behavior of Cheeger deformations on the Ricci curvature. 
	
	Fix a point p ∈ M and consider {e_1,…,e_k,e_k+1,…,e_n} a g-orthonormal base for T_pM such that e_k+1,…,e_n are horizontal and, for i≤ k, e_i = λ_i^-1/2v^∗_i where v_1,…,v_k is a set of Q-orthonormal eigenvectors of P with eigenvalues λ_1,…,λ_k,  taken in non-decreasing order. Observe that {C_t^-1/2e_i}_i=1^n is a g_t-orthonormal base for T_pM. From Proposition <ref>, C_t^-1/2e_i = (1+tλ_i)^1/2e_i for i ≤ k and C_t^-1/2e_i = e_i for i > k.
	
	Define ^ℋ(X) := ∑_i=k+1^nκ_0(X,e_i). We show that _g^ H and the curvature of each orbit (seem as a normal homogeneous space) determine the Ricci curvature. From (<ref>) we have
	
	
    _g_t(C_t^-1X) = ∑_i=1^nR_g_t(C_t^-1/2e_i,C_t^-1X,C_t^-1X,C_t^-1/2e_i) =∑_i=1^nκ_t(C_t^1/2e_i,X)
    
    	= ∑_i=1^nκ_0(C_t^1/2e_i,X) + ∑_i=1^nz_t(C_t^1/2e_i,X) + t^3/4∑_i=1^k[PC_t^1/2λ_i^-1/2v_i,PX]_Q^2
    	
    = _g^ℋ(X) + ∑_i=1^nz_t(C_t^1/2e_i,X) + ∑_i=1^k1/1+tλ_i(κ_0(λ_i^-1/2v^∗_i,X) + λ_it^3/4[v_i,PU]_Q^2 ).

	
	In particular,
	
    _g_t(X) =   _g^ℋ(C_tX) + ∑_i=1^nz_t(C_t^1/2e_i,C_tX) 
       + ∑_i=1^k1/1+tλ_i(κ_0(λ_i^-1/2v^∗_i,C_tX) + λ_it/4[v_i,tP(1+tP)^-1U]_Q^2).

	On the other hand, using Proposition <ref> we get:
	
    lim_t→∞_g^ℋ(C_tX)   =_g^ℋ(X) 
    lim_t→∞∑_i=1^kλ_i/1+tλ_iκ_0(v^∗_i,C_tX)   =0
    lim_t→∞∑_i=1^ktλ_i/4+4tλ_i[v_i,tP(1+tP)^-1U]_Q^2   =∑_i=1^k1/4[v_i,U]_Q^2

	
	
	Consider G/G_p as a normal homogeneous space where G is endowed with the metric defined by Q. Then (<ref>) is exactly the Ricci curvature of G/G_p. In particular, there is a constant K>0 such that ∑_i=1^k1/4[v_i,U]_Q^2≥ K||U||^2_Q, provided G/G_p has finite fundamental group.
	
	With (<ref>)-(<ref>) at hand, a compactness argument shows that g_t has positive Ricci curvature, provided its orbits have finite fundamental group and _g^ H(X)≥ 0 for all non-zero X∈ H.
	
	
		Let (M,g) be a compact Riemannian G-manifold, where G is a compact. Suppose the orbits of M have finite fundamental group and _g^ℋ(X) > 0 for all non-zero  X ∈ℋ. Then g_t has positive Ricci curvature for some t>0. 
	
	
		Define F(t,X)=_g_t(X). We argue by contradiction, supposing that there is no t>0 such that g_t has positive Ricci curvature. Therefore, for each time t=n∈ N, there is a point p_n and a unitary vector X_n=X_n+U_n^* such that F(p_n,t_n,X_n)≤ 0. Passing to a subsequence, if necessary, we conclude that the limit limX_n=X satisfy
		
    0≥lim_n→∞F(t_n,X_n)=_g^ℋ(X)+∑_i=1^k1/4[v_i,U]_Q^2>0,

		a contradiction.
	
	

 §.§ The geometry of cross-diagrams

	We now apply Theorem <ref> to G-G-bundles. Let M ← P → M' be a G-G-bundle. Proposition <ref> guarantees that, given a G-invariant metric g_M on M, there is a G×G-invariant metric g_P on P and (given g_P) a unique metric g_M' on M' that makes π' a submersion.
	
	
	
	
	
	
	
	
	
	
	
	
	Denote by L_π and L_π' the horizontal lifts of π and π', respectively, and  V^π= dπ, V^π'= dπ'. We recall the classical O'Neill formula for both π and π' (see O'Neill <cit.> or Gromoll–Walschap <cit.> for details). Let X,Y∈( V^π)^, X',Y'∈( V^π')^. We have
		
    K_g_M(X,Y)    = K_g_P(ℒ_πX,ℒ_πY) + 3/4[ℒ_πX,ℒ_πY]^𝒱^π^2_g_P,
    		
    K_g_M'(X',Y')    =  K_g_P(ℒ_π'X,ℒ_π'Y) + 3/4[ℒ_π'X,ℒ_π'Y]^𝒱^π'^2_g_P.

	Recall from the proof of Claim <ref> that, given p∈ P, L_p|_ H_π(p): H_π(p)→ H”_p and dπ'|_ H”_p: H”_p→ H'_π'(p) are isometries  (we keep the notation in section <ref>, denoting H, H' and H” the orthogonal to the G-orbits in M, M' and the G×G-orbit on P). Being H” orthogonal to both V^π and V^π',  we get
		
    K_g_M'(dπ'ℒ_πX,dπ'ℒ_πY) =  K_g_M(X,Y) + 3/4{[ℒ_πX,ℒ_πY]^𝒱^π'^2_g_P - [ℒ_πX,ℒ_πY]^𝒱^π^2_g_P}.

	
	We claim we can change the metric on P without affecting the curvatures K_g_M'(dπ'ℒ_πX,dπ'ℒ_πY),   K_g_M(X,Y), but with  [ℒ_πX,ℒ_πY]^𝒱^π^2_g_P arbitrarily small. Precisely:
		
			Let M← P→ M' be a G-G-bundle with P compact. Let g_P be a G×G-invariant metric on P. Then, given ϵ > 0, there exists t > 0 such that the metric g_P_t, obtained by a finite Cheeger deformation on the G×G-manifold P, satisfy: for each pair X,Y∈ H”, 
			
    K_g_M'_t(dπ'X,dπ'Y) -  K_g_M_t(dπ X,dπ Y)≥ -ϵ||X∧ Y||_g_P^2,

			where g_M_t, g_M'_t are the resulting submersion metrics on M and M', respectively.
		
		Observe that, if g_P is G×G-invariant, so it is g_P_t. In fact, the product metric g_P× t^-1Q on P× G is G×G×G-invariant, where G×G×G acts as
		
    (r,s,q)(p,g)=(rpq^-1,sgq^-1).

		Action (<ref>) is given by the q-coordinate, s and r descends to the π-principal and ⋆-actions on the quotient of P× G by q (which is identified with P via (<ref>)). Therefore, the resulting metric g_P_t is invariant both under the π-principal action and the ⋆-action. 
		
		
			It is worth remarking that the resulting metric on M', g_M'_t, is a Cheeger deformation as well.
		
		
		To prove Proposition <ref>, note that the vector [X,Y]^𝒱^π does not change with Cheeger deformation, since  Cheeger deformation does not affect the horizontal (or vertical) space. Therefore, Proposition <ref> follows from Lemma <ref> below. We set some notation first.
		
		
		
		Given p∈ P, let  𝒫_p be the orbit tensor associated to the π-principal action. Given a G× G-invariant metric g_P on P, there exists an orthonormal basis of 𝒫_p-eigenvector    {v_1,...,v_k} for 𝒱_p^π. We denote by λ_i the eigenvalue associated  to v_i. Since the principal action is free and  P is compact, there is positive number λ := min_p∈ P, iλ_i.	
		
		
		Given  ϵ > 0, there exists t > 0 such that g_P_t satisfies, for every V∈ V^π,
		
    g_P_t(V,V) ≤ϵV^2_g_P.

	
	
	 Given p ∈ P, each unitary  V∈ V^π_p  can be written uniquely as V = ∑_i g_P(X,v_i)v_i. In particular,
		
    g_P_t(V,V) = g_P((1 + t𝒫)^-1V,V) = ∑_i,jg_P(V,v_i)g_P(V,v_j)g_P((1+ tλ_i)^-1v_i,v_j).

		Therefore,
		
    g_P_t(V,V) = ∑_ig_P(V,v_i)^21 + tλ_i.

         Once for each i we have that λ_i≥λ we obtain
		
    g_P_t(V,V) = ∑_ig_P(V,v_i)^2/1 + tλ_i≤∑_ig_P(V,v_i)^211+tλ = 11+tλ.
 Since we would like that
		
    11+tλ≤ϵ,

		we must have
		
    t ≥(1-ϵ)λϵ.

	












    	Finally, we recall that the map 
	
    (X,Y)↦[X,Y]^𝒱^π^2_g_P

	is tensorial on X,Y∈ (V^π)^ (see, for instance, the definition of the A-tensor on <cit.> or <cit.>). Therefore, assuming P compact, there is a constant C>0 such that   
    [X,Y]^𝒱^π^2_g_P≤ C||X||_g_P^2||Y||^2_g_P.

    Observing that g_P_t induces the same metric g_M for every t, we conclude that, for any given ϵ>0 and any G-invariant metric g_M, there is a metric g_M' such that
    
    K_g_M'(dπ'X,dπ'Y) =  K_g_M(dπ X,dπ Y) + 3/4{[X,Y]^𝒱^π'^2_g_P - [X,Y]^𝒱^π^2_g_P}
    ≥  K_g_M(dπ X,dπ Y) -ϵ||X∧ Y||_g_P^2.


		   
	Taking {e_1, …, e_k} an orthonormal base for ℋ_p” and X∈ H_π'(p),  we have:
	
    ^ℋ_g_M'(X)  ≥^ℋ_g_M(dπℒ_π'X)-ϵ||X||_g_M'^2.

	
	Recalling that  M'← P→ M is a G-G-bundle as well, we conclude:
	
	
	Let M'← P→ M be a G-G-bundle ans suppose P compact. Then, M has a G-invariant metric of positive horizontal Ricci curvature if and only if M' does so.
	
	
	
	
	Positive Ricci curvature on the exotic projective planes Σ P C^2m-1, Σ P H^2m-1 follows from Theorem <ref> since vectors orthogonal to the orbits of the U(m)- and Sp(m)-action lift to vectors orthogonal to the O(n)-orbits on Σ^2n-1. In fact, positive horizontal Ricci curvature on Σ^2n-1 (provided by Theorem <ref>) ensures positive horizontal Ricci curvature on  Σ P C^2m-1 and Σ P H^2m-1. The conclusion follows from the fact the U(m)-, respectively, Sp(m)-orbits have finite fundamental group.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
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