Cartan–Ambrose–Hicks theorem

In mathematics, the Cartan–Ambrose–Hicks theorem is a theorem of Riemannian geometry, according to which the Riemannian metric is locally determined by the Riemann curvature tensor, or in other words, behavior of the curvature tensor under parallel translation determines the metric.

The theorem is named after Élie Cartan, Warren Ambrose, and his PhD student Noel Hicks.[1] Cartan proved the local version. Ambrose proved a global version that allows for isometries between general Riemannian manifolds with varying curvature, in 1956.[2] This was further generalized by Hicks to general manifolds with affine connections in their tangent bundles, in 1959.[3]

A statement and proof of the theorem can be found in [4]

Introduction

Let be connected, complete Riemannian manifolds. Let , and let

be a linear isometry. For sufficiently small , the exponential maps

are local diffeomorphisms. Here, is the ball centered on of radius One then defines a diffeomorphism by

For a geodesic with , maps it to a geodesic with ,. Let be the parallel transport along (defined by the Levi-Civita connection), and be the parallel transport along . We then define

for .

Cartan's theorem

The original theorem proven by Cartan is the local version of the Cartan–Ambrose–Hicks theorem. It states that is a (local) isometry if for all geodesics with and all , we have , where are Riemann curvature tensors of .

Note that generally does not have to be a diffeomorphism, but only a locally isometric covering map. However, must be a global isometry if is simply connected.

Cartan–Ambrose–Hicks theorem

Theorem: For Riemann curvature tensors and all broken geodesics (a broken geodesic is a curve that is piecewise geodesic) with ,

for all .

Then, if two broken geodesics beginning in have the same endpoint, then the corresponding broken geodesics (mapped by ) in also have the same end point. So there exists a map

by mapping the broken geodesic endpoints in to the corresponding geodesic endpoints in .

The map is a locally isometric covering map.

If is also simply connected, then is an isometry.

Locally symmetric spaces

A Riemannian manifold is called locally symmetric if its Riemann curvature tensor is invariant under parallel transport:

A simply connected Riemannian manifold is locally symmetric if it is a symmetric space.

From the Cartan–Ambrose–Hicks theorem, we have:

Theorem: Let be connected, complete, locally symmetric Riemannian manifolds, and let be simply connected. Let their Riemann curvature tensors be . Let and

be a linear isometry with . Then there exists a locally isometric covering map

with and .

Corollary: Any complete locally symmetric space is of the form for a symmetric space and is a discrete subgroup of isometries of .

Classification of space forms

As an application of the Cartan–Ambrose–Hicks theorem, any simply connected, complete Riemannian manifold with constant sectional curvature is respectively isometric to the n-sphere , the n-Euclidean space , and the n-hyperbolic space .

References

  1. Mathematics Genealogy Project, entry for Noel Justin Hicks
  2. Ambrose, W. (1956). "Parallel Translation of Riemannian Curvature". The Annals of Mathematics. JSTOR. 64 (2): 337. doi:10.2307/1969978. ISSN 0003-486X.
  3. Hicks, Noel (1959). "A theorem on affine connexions". Illinois Journal of Mathematics. 3 (2): 242–254. doi:10.1215/ijm/1255455125. ISSN 0019-2082.
  4. Cheeger, Jeff; Ebin, David G. (2008). "Chapter 1, Section 12, The Cartan–Ambrose–Hicks Theorem". Comparison theorems in Riemannian geometry. Providence, R.I: AMS Chelsea Pub. ISBN 0-8218-4417-2. OCLC 185095562.
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