Jurgen Berndt

Geodesics in weakly symmetric spaces

We prove that any maximal geodesic in a weakly symmetric space is an orbit of a one-parameter group of isometries of that space.

Two natural generalizations of locally symmetric spaces

Abstract One studies two classes of Riemannian manifolds which extend the class of locally symmetric spaces: manifolds all of whose Jacobi operators Rγ have constant eigenvalues ( C -spaces) or parallel eigenspaces ( B -spaces) along geodesics γ. One gives several examples, derives equivalent characterizations and treats classifications for the two- and the three-dimensional case.

Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperbolic Spaces

We present the classification of all real hypersurfaces in complex hyperbolic space

HYPERSURFACES IN NONCOMPACT COMPLEX GRASSMANNIANS OF RANK TWO

\mathbb{C}H^{n}

SYMMETRIC-LIKE RIEMANNIAN MANIFOLDS AND GEODESIC SYMMETRIES

,

REAL HYPERSURFACES WITH ISOMETRIC REEB FLOW IN COMPLEX QUADRICS

n \geq 3

Weakly symmetric groups of Heisenberg type

, with three distinct constant principal curvatures

Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

Consider a Riemannian manifold N equipped with an additional geometric structure, such as a Kahler structure or a quaternionic Kahler structure, and a hypersurface M in N. The geometric structure induces a decomposition of the tangent bundle TM of M into subbundles. A natural problem is to classify all hypersurfaces in N for which the second fundamental form of M preserves these subbundles. This problem is reasonably well understood for Riemannian symmetric spaces of rank one, but not for higher rank symmetric spaces. A general treatment of this problem for higher rank symmetric spaces is out of reach at present, and therefore it is desirable to understand this problem better in a few special cases. Due to some conceptual differences between symmetric spaces of compact type and of noncompact type it appears that one needs to consider these two cases separately. In this paper we investigate this problem for the rank two symmetric space SU2, m/S(U2Um) of noncompact type.

Contact hypersurfaces in Kähler manifolds

We treat several classes of Riemannian manifolds whose shape operators of geodesic spheres or Jacobi operators share some properties with the ones on symmetric spaces.

Almost complex curves and Hopf hypersurfaces in the nearly Kähler 6-sphere

We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Q^m for m > 2. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space CP^k which is embedded canonically in Q^{2k} as a totally geodesic complex submanifold. As a consequence we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics.