Author
Jurgen Berndt
Other affiliations: University of Cologne, Stanford University, Katholieke Universiteit Leuven ...read more
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.
Papers published on a yearly basis
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TL;DR: In this paper, the authors generalize this result in two directions by considering the projective planes over the normed real division algebras and considering the complexifications of these four projective plane complexifications.
Abstract: A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes.
6 citations
Journal Article•
6 citations
Journal Article•
5 citations
Posted Content•
TL;DR: In this paper, a systematic approach to the construction and classification of homogeneous submanifolds whose principal curvatures are independent of the normal direction in irreducible Riemannian symmetric spaces of non-compact type and rank greater than one is presented.
Abstract: We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds, homogeneous austere hypersurfaces, and singular orbits of cohomogeneity one actions. The main purpose of this article is to present a systematic approach to the construction and classification of homogeneous submanifolds whose principal curvatures are independent of the normal direction in irreducible Riemannian symmetric spaces of non-compact type and rank greater than one.
5 citations
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01 Jan 2015
TL;DR: 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 hypersurface in spheres as discussed by the authors.
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.
228 citations
TL;DR: In this paper, the Ricci soliton left invariant Riemannian metric is shown to be unique up to isometry and scaling, if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifold is Einstein.
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.
220 citations
Book•
01 Jan 2007TL;DR: The theory of commutative spaces has been studied in the context of topological group theory as discussed by the authors, where spherical transforms and Plancherel formulae have been used.
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.
203 citations
189 citations
Journal Article•
TL;DR: In this paper, 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.
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 ∫
178 citations