Rigidity

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As rank rigidity (engl .: rank rigidity ) is referred to in the differential geometry , the phenomenon that only for very special Riemannian manifolds rank is greater than the first

These examples of higher rank essentially include locally symmetric spaces of the corresponding rank as well as manifolds whose universal superposition is a Riemannian product .

Non-positive curvature

If the rank is greater than 1 for a Riemannian manifold with non-positive sectional curvature , then it is either locally symmetric or its universal superposition is a Riemannian product.

An analogous property also applies to CAT (0) cube complexes .

Non-negative curvature

There are examples of 9-dimensional manifolds of nonnegative section curvature, which have higher rank, but cannot be homotopy equivalent to a compact , locally homogeneous space and whose universal superposition cannot be decomposed as a Riemannian product.

Curvature of variable sign

For a 3-dimensional Riemannian manifold with a rank greater than 1, the universal covering must be a Riemannian product.

Individual evidence

  1. ↑ Werner Ballmann : Nonpositively curved manifolds of higher rank , Ann. Math. 122: 597-609 (1985)
  2. ↑ Keith Burns , Ralf Spatzier : Manifolds of nonpositive curvature and their buildings , Publ. IHES 65, 35-59 (1987)
  3. ↑ Patrick Eberlein , Jens Heber : A differential geometric characterization of symmetric spaces of higher rank , Publ. IHES 71, 33-44 (1990)
  4. ↑ Pierre-Emmanuel Caprace , Michah Sageev : Rank rigidity for CAT (0) cube complexes , GAFA 21 (2011)
  5. ^ R. Spatzier, M. Strake: Some examples of higher rank manifolds of nonnegative curvature , Comm. Math. Helv. 65, 299-317 (1990)
  6. ^ R. Bettiol, B. Schmidt: Three-manifolds with many flat planes , Trans. AMS 370, 669-693 (2018)