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Ernst Heintze

Publications and source records attributed to Ernst Heintze.

7 recordsLinked to original sources
arXiv2025

Symmetric Spaces with Rectangular Unit Lattices, Revisited

We give a new proof of a theorem of Loos stating that a Riemannian symmetric space X with rectangular unit lattice is a symmetric R-space. For this we construct explicitly an isometric extrinsically symmetric embedding of X in a Euclidean space which reveals X as a standardly embedded symmetric R-space. We further determine the root systems, Euclidean root data, fundamental groups and eigenvalues of the Laplacian of symmetric spaces with rectangular unit lattice in a direct way.

math.DG↗
arXiv2012

Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space

We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by a result of Heintze and Liu, and associate to such a submanifold M and a point x in M a canonical homogeneous structure (a certain bilinear map on the tangent space). We prove that the homogeneous structure together with the second fundamental form encodes all the information about M, and deduce from this the rigidity result that M is completely determined by the second fundamental form and its covariant derivative, thereby making such submanifolds accessible to classification. As an essential step, we show that the one-parameter groups of isometries constructed by Heintze and Liu to prove their homogeneity induce smooth and hence everywhere defined Killing fields, implying the continuity of the homogeneous structure. Here an important tool is the introduction of affine root systems of isoparametric submanifolds.

math.DG↗
arXiv2009

Finite order automorphisms and real forms of affine Kac-Moody algebras in the smooth and algebraic category

Automorphisms of finite order and real forms of "smooth" affine Kac-Moody algebras are studied, i.e. of 2-dimensional extensions of the algebra of smooth loops in a simple Lie algebra. It is shown that they can be parametrized by certain invariants and that in particular the classification of involutions essentially follows from Cartan's classifications in finite dimensions. We also prove that our approach works equally well in the usual algebraic setting and leads to the same results there.

math.RA↗
arXiv2000

Isoparametric submanifolds and a Chevalley-type restriction theorem

We define and study isoparametric submanifolds of general ambient spaces and of arbitrary codimension. In particular we study their behaviour with respect to Riemannian submersions and their lift into a Hilbert space. These results are used to prove a Chevalley type restriction theorem which relates by restriction eigenfunctions of the Laplacian on a compact Riemannian manifold which contains an isoparametric submanifold with flat sections to eigenfunctions of the Laplacian of a section. A simple example of such an isoparametric foliation is given by the conjugacy classes of a compact Lie group and in that case the restriction theorem is a (well known) fundamental result in representation theory. As an application of the restriction theorem we show that isoparametric submanifolds with flat sections in compact symmetric spaces are level sets of eigenfunctions of the Laplacian and are hence related to representation theory. In addition we also get the following results. Isoparametric submanifolds in Hilbert space have globally flat normal bundle, and a general result about Riemannian submersions which says that focal distances do not change if a submanifold of the base is lifted to the total space.

math.DG↗
arXiv1999

Homogeneity of infinite dimensional isoparametric submanifolds

A subset S of a Riemannian manifold N is called extrinsically homogeneous if S is an orbit of a subgroup of the isometry group of N. Thorbergsson proved the remarkable result that every complete, connected, full, irreducible isoparametric submanifold of a finite dimensional Euclidean space of rank at least 3 is extrinsically homogeneous. This result, combined with results of Palais-Terng and Dadok, finally classified irreducible isoparametric submanifolds of a finite dimensional Euclidean space of rank at least 3. While Thorbergsson's proof used Tits buildings, a simpler proof without using Tits buildings was given by Olmos. The main purpose of this paper is to extend Thorbergsson's result to the infinite dimensional case.

math.DG↗