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.

Cohomogeneity one actions on symmetric spaces of noncompact type

Abstract: An isometric action of a Lie group on a Riemannian manifold is of cohomogeneity one if the corresponding orbit space is one-dimensional. In this article we develop a conceptual approach to the classification of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type in terms of orbit equivalence. As a consequence, we find many new examples of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type. We apply our conceptual approach to derive explicit classifications of cohomogeneity one actions on some symmetric spaces.

Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

Abstract: We classify real hypersurfaces with constant principal curvatures in the complex hyperbolic plane. It follows from this classification that all of them are open parts of homogeneous ones. © 2007 American Mathematical Society.

Hyperpolar homogeneous foliations on symmetric spaces of noncompact type

Abstract: A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M . A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F , and intersects orthogonally at each point of intersection. A foliation F is hyperpolar if it admits a flat section. These notes are related to joint work with Jose Carlos Diaz-Ramos and Hiroshi Tamaru about hyperpolar homogeneous foliations on Riemannian symmetric spaces of noncompact type. Apart from the classification result which we proved in [1], we present here in more detail some relevant material about symmetric spaces of noncompact type, and discuss the classification in more detail for the special case M = SLr+1(R)/SOr+1. 1 Isometric actions on the real hyperbolic plane The special linear group G = SL2(R) = {( a b c d )∣∣∣∣ a, b, c, d ∈ R, ad− bc = 1} acts on the upper half plane H = {z ∈ C | =(z) > 0} by linear fractional transformations of the form ( a b c d ) · z = az + b cz + d . By equipping H with the Riemannian metric ds = dx + dy y2 = dzdz =(z)2 (z = x+ iy)

Homogeneous hypersurfaces in complex hyperbolic spaces

Abstract: We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.

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 ∫