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. Cartan proved the local version. Ambrose proved a global version that allows for isometries between general Riemannian manifolds with varying curvature, in 1956. This was further generalized by Hicks to general manifolds with affine connections in their tangent bundles, in 1959.
A statement and proof of the theorem can be found in

Introduction

Let be connected, complete Riemannian manifolds. We consider the problem of isometrically mapping a small patch on to a small patch on.
Let, and let
be a linear isometry. This can be interpreted as isometrically mapping an infinitesimal patch at to an infinitesimal patch at. Now we attempt to extend it to a finite patch.
For sufficiently small, the exponential maps
are local diffeomorphisms. Here, is the ball centered on of radius One then defines a diffeomorphism by
When is an isometry? Intuitively, it should be an isometry if it satisfies the two conditions:
  • It is a linear isometry at the tangent space of every point on, that is, it is an isometry on the infinitesimal patches.
  • It preserves the curvature tensor at the tangent space of every point on, that is, it preserves how the infinitesimal patches fit together.
If is an isometry, it must preserve the geodesics. Thus, it is natural to consider the behavior of as we transport it along an arbitrary geodesic radius starting at. By property of the exponential mapping, maps it to a geodesic radius of starting at,.
Let be the parallel transport along, and be the parallel transport along, then we have the mapping between infinitesimal patches along the two geodesic radii:

Cartan's theorem

The original theorem proven by Cartan is the local version of the Cartan–Ambrose–Hicks theorem.
is an isometry if and only if for all geodesic radii with, and all, we have
where are Riemann curvature tensors of.
In words, it states that is an isometry if and only if the only way to preserve its infinitesimal isometry also preserves the Riemannian curvature.
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 with, suppose that
for all.
Then, if two broken geodesics beginning at have the same endpoint, the corresponding broken geodesics in also have the same end point. Consequently, there exists a map defined
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, where is a symmetric space and is a discrete subgroup of isometries of.

Classification of [space form]s

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.