Locally symmetrical space

In mathematics , locally symmetric spaces are an important class of examples in differential geometry , particularly including flat manifolds and hyperbolic manifolds . Harmonic analysis on locally symmetric spaces is closely related to the theory of automorphic forms and has deep-seated applications in number theory .

properties

A Riemannian manifold is a locally symmetric space if it fulfills one of the following equivalent properties: ${\ displaystyle M}$

For each there is a , so that the geodetic reflection is an isometry of the sphere on itself. ${\ displaystyle x \ in M}$ ${\ displaystyle \ epsilon> 0}$ ${\ displaystyle \ epsilon}$ ${\ displaystyle B _ {\ epsilon} (x)}$

The derivation of the Riemann curvature tensor vanishes:

{\ displaystyle \ nabla R = 0}

.

For each Jacobi field with is .

{\ displaystyle Y (0) = 0}

{\ displaystyle \ | Y (t) \ | = \ | Y (-t) \ |}

The universal overlay is a symmetrical space . ${\ displaystyle {\ widetilde {M}}}$

{\ displaystyle \ Gamma \ subset \ operatorname {isom} ({\ widetilde {M}})}

{\ displaystyle M = \ Gamma \ backslash {\ widetilde {M}}}

.

Flat manifolds are of the form for a lattice and are therefore locally symmetrical spaces. ${\ displaystyle \ Gamma \ backslash \ mathbb {R} ^ {n}}$ ${\ displaystyle \ Gamma \ subset \ mathbb {R} ^ {n}}$

If is a semisimple Lie group , a maximally compact subgroup, and a discrete torsion-free subgroup, then is a locally symmetric space. ${\ displaystyle G}$ ${\ displaystyle K \ subset G}$ ${\ displaystyle \ Gamma \ subset G}$ ${\ displaystyle \ Gamma \ backslash G / K}$

For and is the hyperbolic space , in particular hyperbolic manifolds are examples of locally symmetric spaces. ${\ displaystyle G = SO_ {0} (n, 1)}$ ${\ displaystyle K = SO (n)}$ ${\ displaystyle G / K}$ ${\ displaystyle \ Gamma \ backslash H ^ {n}}$

The locally symmetric spaces are important for number theory and the theory of automorphic forms .

{\ displaystyle SL (n, \ mathbb {Z}) \ backslash SL (n, \ mathbb {R}) / SO (n)}