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 .
The rank in is the maximum number of linearly independent , parallel Jacobi fields along geodesics through .
The rank of is the minimum of the rank in all points of .
Rigidity
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