Jurgen Berndt

Bio: Jurgen Berndt is an academic researcher from King's College London. The author has contributed to research in topics: Symmetric space & Submanifold. The author has an hindex of 26, co-authored 84 publications receiving 2343 citations. Previous affiliations of Jurgen Berndt include University of Cologne & Stanford University.

Totally Geodesic Submanifolds of Riemannian Symmetric Spaces

Abstract: The index of a Riemannian manifold is defined as the minimal codimension of a totally geodesic submanifold. In this note we discuss two recent results by the author and Olmos (Berndt and Olmos, On the index of symmetric spaces, preprint, arXiv:1401.3585) and some related topics. The first result states that the index of an irreducible Riemannian symmetric space is bounded from below by the rank of the symmetric space. The second result is the classification of all irreducible Riemannian symmetric spaces of noncompact type whose index is less or equal than three.

The index of compact simple Lie groups

Abstract: Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a (non-trivial) totally geodesic submanifold of M. The purpose of this note is to determine the index i(M) for all irreducible Riemannian symmetric spaces M of type (II) and (IV).

Maximal totally geodesic submanifolds and index of symmetric spaces

Abstract: Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In previous work the authors proved that i(M) is bounded from below by the rank rk(M) of M. In this paper we classify all irreducible Riemannian symmetric spaces M for which the equality holds, that is, rk(M) = i(M). In this context we also obtain an explicit classification of all non-semisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric R-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with i(M) = 4,5 or 6.

Real Hypersurfaces in Complex Space Forms

Abstract: The study of real hypersurfaces in complex projective space CP n and complex hyperbolic space CH n began at approximately the same time as Munzner’s work on isoparametric hypersurfaces in spheres. A key early work was Takagi’s classification [669] in 1973 of homogeneous real hypersurfaces in CP n . These hypersurfaces necessarily have constant principal curvatures, and they serve as model spaces for many subsequent classification theorems. Later Montiel [501] provided a similar list of standard examples in complex hyperbolic space CH n . In this chapter, we describe these examples of Takagi and Montiel in detail, and later we prove many important classification results involving them. We also study Hopf hypersurfaces, focal sets, parallel hypersurfaces and tubes using both standard techniques of submanifold geometry and the method of Jacobi fields.

Ricci soliton homogeneous nilmanifolds

Abstract: We study a notion weakening the Einstein condition on a left invariant Riemannian metric g on a nilpotent Lie groupN. We consider those metrics satisfying Ric $_g=cI+D$ for some $c\in{mathbb R}$ and some derivationD of the Lie algebra ${\mathfrak n}$ ofN, where Ric $_g$ denotes the Ricci operator of $(N,g)$ . This condition is equivalent to the metric g to be a Ricci soliton. We prove that a Ricci soliton left invariant metric on N is unique up to isometry and scaling. The following characterization is also given: (N,g) is a Ricci soliton if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifold $(S,\tilde{g})$ is Einstein. This gives several families of new examples of Ricci solitons. By a variational approach, we furthermore show that the Ricci soliton homogeneous nilmanifolds (N,g) are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set.

Harmonic analysis on commutative spaces

Abstract: General theory of topological groups: Basic topological group theory Some examples Integration and convolution Representation theory and compact groups: Basic representation theory Representations of compact groups Compact Lie groups and homogeneous spaces Discrete co-compact subgroups Introduction to commutative spaces: Basic theory of commutative spaces Spherical transforms and Plancherel formulae Special case: Commutative groups Structure and analysis for commutative spaces: Riemannian symmetric spaces Weakly symmetric and reductive commutative spaces Structure of commutative nilmanifolds Analysis on commutative nilmanifolds Classification of commutative spaces Bibliography Subject index Symbol index Table index.

A short survey on biharmonic maps between Riemannian manifolds

Abstract: and the corresponding Euler-Lagrange equation is H = 0, where H is the mean curvature vector field. If φ : (M, g) → (N, h) is a Riemannian immersion, then it is a critical point of the bienergy in C∞(M,N) if and only if it is a minimal immersion [26]. Thus, in order to study minimal immersions one can look at harmonic Riemannian immersions. A natural generalization of harmonic maps and minimal immersions can be given by considering the functionals obtained integrating the square of the norm of the tension field or of the mean curvature vector field, respectively. More precisely: • biharmonic maps are the critical points of the bienergy functional E2 : C∞(M,N) → R, E2(φ) = 12 ∫