Cartan-Ambrose-Hicks theorem

In mathematics , the Cartan-Ambrose-Hicks theorem is a theorem of Riemannian geometry , according to which Riemannian metrics are locally uniquely defined by the Riemannian curvature tensor .

The set is named after Élie Cartan , who proved the local version, and Warren Ambrose and his graduate student Noel Hicks. Ambrose proved a global version in 1956.

Preparations

Let be connected, complete Riemannian manifolds, and ${\displaystyle M,N}$ ${\displaystyle x\in M,y\in N}$

${\displaystyle I:T_{x}M\rightarrow T_{y}N}$

a linear isometry. The exponential maps are for sufficiently small ${\displaystyle r>0}$

${\displaystyle \exp _{x}:B_{r}(x)\subset T_{x}M\rightarrow M,\exp _{y}:B_{r}(y)\subset T_{y}N\rightarrow N}$

local diffeomorphisms. A differentiable mapping is then defined by ${\displaystyle f:B_{r}(x)\rightarrow B_{r}(y)}$

${\displaystyle f=\exp _{y}\circ I\circ \exp _{x}^{-1}}$.

For a geodesic with be the parallel transport (defined by means of the Levi-Civita connection ) along . We then define ${\displaystyle \gamma :\left[0,T\right]\rightarrow B_{r}(x)\subset M}$ ${\displaystyle \gamma (0)=x}$ ${\displaystyle P_{\gamma }}$ ${\displaystyle \gamma }$

${\displaystyle I_{\gamma }=P_{f(\gamma )}\circ I\circ P_{\gamma }^{-1}:T_{\gamma (t)}M\rightarrow T_{f(\gamma (t))}N}$

for . ${\displaystyle t\in \left[0,T\right]}$

Cartan's theorem

The original Cartan Theorem is the local version of the Cartan-Ambrose-Hicks Theorem. It says that there is a (local) isometry if and only if the following applies to all geodesics with and all : ${\displaystyle f}$ ${\displaystyle \gamma :\left[0,T\right]\rightarrow B_{r}(x)\subset M}$ ${\displaystyle \gamma (0)=x}$ ${\displaystyle X,Y,Z\in T_{\gamma (T)}M}$

${\displaystyle I_{\gamma }(R_{XY}Z)={\overline {R}}_{I_{\gamma }XI_{\gamma }Y}(I_{\gamma }Z)}$,

where are the Riemann curvature tensors of . ${\displaystyle R,{\overline {R}}}$ ${\displaystyle M,N}$

Note that there is generally no need for a diffeomorphism, but only a local isometric overlay . However, it must be a global isometric drawing if it is simply connected. ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle N}$

Cartan-Ambrose-Hicks theorem

Let be connected, complete Riemannian manifolds, simply connected. Be and ${\displaystyle M,N}$ ${\displaystyle M}$ ${\displaystyle x\in M,y\in N}$

${\displaystyle I:T_{x}M\rightarrow T_{y}N}$

a linear isometry. For the Riemann curvature tensors and all in incipient broken geodesic applies ${\displaystyle R,{\overline {R}}}$ ${\displaystyle x}$ ${\displaystyle \gamma :\left[0,T\right]\rightarrow M}$

${\displaystyle I_{\gamma }(R_{XY}Z)={\overline {R}}_{I_{\gamma }XI_{\gamma }Y}(I_{\gamma }Z)}$

for everyone . ${\displaystyle X,Y,Z\in T_{\gamma (T)}M}$

Then the following applies: if two broken geodesics beginning in have the same end point, then this also applies to the (sub ) corresponding broken geodesics in . So you can make a figure ${\displaystyle x}$ ${\displaystyle I_{\gamma }}$ ${\displaystyle N}$

${\displaystyle F:M\rightarrow N}$

by mapping the endpoints of fractional geodesics to the endpoints of the corresponding geodesics in . ${\displaystyle N}$

The figure is a local isometric overlay. ${\displaystyle F:M\rightarrow N}$

If is also simply connected, then is an isometry. ${\displaystyle N}$ ${\displaystyle F}$

Locally symmetrical spaces

A Riemann manifold is called locally symmetric if the Riemann curvature tensor is parallel:

${\displaystyle \nabla R=0}$.

A simply connected Riemann manifold is locally symmetric if and only if it is a symmetric space .

The Cartan-Ambrose-Hicks theorem gives:

Theorem : Let be connected, complete, locally symmetric Riemannian manifolds, simply connected. Be and ${\displaystyle M,N}$ ${\displaystyle M}$ ${\displaystyle x\in M,y\in N}$

${\displaystyle I:T_{x}M\rightarrow T_{y}N}$

a linear isometry with

${\displaystyle I(R_{XY}Z)={\overline {R}}_{I(X)I(Y)}I(Z)}$

for the Riemann curvature tensors . Then there is a locally isometric overlay ${\displaystyle R,{\overline {R}}}$

${\displaystyle F:M\rightarrow N}$

with and . ${\displaystyle F(x)=y}$ ${\displaystyle D_{x}F=I}$

As a corollary it follows that every complete locally symmetric space is of the form for a symmetric space and a discrete group of isometries . ${\displaystyle \Gamma \backslash M}$ ${\displaystyle M}$ ${\displaystyle \Gamma \subset Isom(M)}$

Spatial forms

As an application of the Cartan-Ambrose-Hicks theorem, every simply connected, complete Riemannian manifold with constant sectional curvature is isometric to the standard sphere or the Euclidean space or the hyperbolic space . ${\displaystyle \in \left\{+1,0,-1\right\}}$ ${\displaystyle S^{n}}$ ${\displaystyle E^{n}}$ ${\displaystyle \mathbb {H} ^{n}}$

The following also applies:

• every complete Riemannian manifold with constant sectional curvature is of the form for a finite group of isometries , ${\displaystyle +1}$ ${\displaystyle \Gamma \backslash S^{n}}$ ${\displaystyle \Gamma \subset Isom(S^{n})}$
• every complete Riemannian manifold with constant cutting curvature is of the form for a Bieberbach group , ${\displaystyle 0}$ ${\displaystyle \Gamma \backslash E^{n}}$ ${\displaystyle \Gamma \subset Isom(E^{n})}$
• every complete Riemannian manifold with constant section curvature is of the form for a discrete group of isometries . ${\displaystyle -1}$ ${\displaystyle \Gamma \backslash \mathbb {H} ^{n}}$ ${\displaystyle \Gamma \subset Isom(\mathbb {H} ^{n})}$

literature

• Jeff Cheeger , David Ebin: Comparison theorems in Riemannian geometry. Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. ISBN 978-0-8218-4417-5
• Joseph A. Wolf : Spaces of constant curvature. Sixth edition. AMS Chelsea Publishing, Providence, RI, 2011. ISBN 978-0-8218-5282-8
• Fangyang Zheng: Complex differential geometry. AMS / IP Studies in Advanced Mathematics, 18th American Mathematical Society, Providence, RI; International Press, Boston, MA, 2000. ISBN 0-8218-2163-6

Individual evidence

1. ^ Mathematics Genealogy Project , entry on N. Hicks
2. ^ W. Ambrose: Parallel translation of Riemannian curvature , Annals of Mathematics (2) 64 (1956), 337-363