Rank (differential geometry)

In the mathematical field of differential geometry , rank measures the presence of flax in a Riemannian manifold.

definition

Let be a Riemannian manifold and . ${\displaystyle M}$ ${\displaystyle p\in M}$

The rank in is the maximum number of linearly independent , parallel Jacobi fields along geodesics through . ${\displaystyle p}$ ${\displaystyle p}$

The rank of is the minimum of the rank in all points of . ${\displaystyle M}$ ${\displaystyle M}$

Rigidity

→ Main article : Rigidity of rank

The phenomenon of rigidity of rank means that the rank is greater than 1 only for very special Riemannian manifolds.

literature

• Werner Ballmann , Michael Gromov , Viktor Schroeder : Manifolds of nonpositive curvature , Birkhäuser, ISBN 978-1-4684-9161-6