1509.02125.pdf - LINKING CURVES SUTURED MANIFOLDS AND THE...

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LINKING CURVES, SUTURED MANIFOLDS AND THE AMBROSE CONJECTURE FOR GENERIC 3-MANIFOLDS PABLO ANGULO ARDOY Abstract. We present a new strategy for proving the Ambrose conjecture , a global version of the Cartan local lemma. We introduce the concepts of linking curves, unequivocal sets and sutured manifolds, and show that any sutured manifold satisfies the Ambrose conjecture. We then prove that the set of sutured Riemannian manifolds contains a residual set of the metrics on a given smooth manifold of dimension 3 . 1. Introduction Let ( M 1 , g 1 ) and ( M 2 , g 2 ) be two complete Riemannian manifolds of the same di- mension , with selected points p 1 M 1 and p 2 M 2 . Any linear map L : T p 1 M 1 T p 2 M 2 induces a map between the pointed manifolds ( M 1 , p 1 ) and ( M 2 , p 2 ): ϕ = exp p 2 L (exp p 1 | O 1 ) - 1 defined in ϕ ( O 1 ) , for any domain O 1 T p 1 M 1 such that e 1 | O 1 is injective (for example, if exp p 1 O 1 is a normal neighborhood of p 1 ). A classical theorem of E. Cartan [C] identifies a situation where this map is an isometry. For x T p 1 M 1 , let γ 1 be the geodesic on M 1 defined in the interval [0 , 1] , starting at p 1 with initial speed vector x and γ 2 be the geodesic on M 2 starting at p 2 with initial speed L ( x ) . Let P γ : T p i M i T γ i (1) M i denote parallel transport along a curve γ . Definition 1.1. The curvature tensors of ( M 1 , p 1 ) and ( M 2 , p 2 ) are L -related if and only if for any x T p 1 M 1 : (1.1) P * γ 1 ( R γ 1 (1) ) = L * P * γ 2 ( R γ 2 (1) ) In the definition, P * γ i ( R γ i (1) ) is the pull back of the (0 , 4) curvature tensor at γ i (1) M i by the linear isometry P γ i , for i = 1 , 2 P * γ i ( R γ i (1) )( v 1 , v 2 , v 3 , v 4 ) = R γ i (1) ( P γ i ( v 1 ) , P γ i ( v 2 ) , P γ i ( v 3 ) , P γ i ( v 4 ) ) for any four vectors v 1 , v 2 , v 3 , v 4 in T p i M i , and L * is used to carry the tensor P * γ 2 ( R γ 2 (1) ) from p 2 M 2 to p 1 M 1 . The usual way to express that the curvature tensors of ( M 1 , p 1 ) and ( M 2 , p 2 ) are L -related is to say that the parallel translation of curvature along corresponding geodesics on M 1 and M 2 coincides. This certainly holds if L is the differential of a global isometry between M 1 and M 2 . Theorem 1.2. If the curvature tensors of ( M 1 , p 1 ) and ( M 2 , p 2 ) are L -related, and exp 1 | O 1 is injective for some domain O 1 T p 1 M 1 , then ϕ = exp 2 L (exp 1 | O 1 ) - 1 is an isometric immersion. Proof. The proof of lemma 1.35 in [CE] works for any domain O such that exp 1 | O is injective. The author was partially supported by research grant ERC 301179, and by INEM. 1 arXiv:1509.02125v2 [math.DG] 4 Apr 2016
Pablo Angulo-Ardoy In 1956 (see [A]), W. Ambrose proved a global version of the above theorem, but with stronger hypothesis. A broken geodesic is the concatenation of a finite amount of geodesic segments. The Ambrose’s theorem states that if the parallel translation of curvature along broken geodesics on M 1 and M 2 coincide, and both manifolds are simply connected , then the above construction gives a global isometry ϕ : M 1 M 2 whose differential at p 1 is L . It is enough if the hypothesis holds for broken geodesics with only one elbow ” (the reader can find more details in [CE]). In [Hi], in 1959, the result of W. Ambrose was generalized to parallel transport
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