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 (the tangent space) at  to an infinitesimal patch at . Now we attempt to extend it to a finite (rather than infinitesimal) 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  (defined by the Levi-Civita connection), 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 (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 

Riemannian geometry