arXiv subjects
Publications and source records attributed to Tillmann Jentsch.
Every homogeneous Riemannian C_0-space (N,g) is associated with its minimal polynomial. To provide explicit examples, we compute the minimal polynomials for generalized Heisenberg groups equipped with their canonical left-invariant metrics.
We construct, at each point of a Riemannian C_0-space, a polynomial in one variable whose coefficients are polynomial functions on the tangent space. For a real-analytic Riemannian C_0-space these pointwise-defined polynomials glue together to a global polynomial whose coefficients are Killing tensors invariant under the full isometry group. Moreover, the degree of this polynomial provides an upper bound for the Singer invariant of the space.
Naturally reductive spaces, in general, can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless, there are some that are closer to symmetric spaces than others. On the one hand, there is the series of Hopf fibrations over complex space forms, including the Heisenberg groups with their metrics of type H. On the other hand, there exist certain naturally reductive spaces in dimensions six and seven whose torsion forms have a distinguished algebraic property. All these spaces generalize geometric or algebraic properties of $3$--dimensional naturally reductive spaces and have the following point in common: along every geodesic the Jacobi operator satisfies an ordinary differential equation with constant coefficients which can be chosen independently of the given geodesic.
We study symmetric Killing 2-tensors on Riemannian manifolds and show that several additional conditions can be realised only for Sasakian manifolds and Euclidean spheres. In particular we show that (three)-Sasakian manifolds can also be characterized by properties of the symmetric products of their characteristic 1-forms. Moreover, we recover a result of S.~Gallot on the characterization of spheres by means of functions satisfying a certain differential equation of order three.
In every point of a Kähler manifold there exist special holomorphic coordinates well adapted to the underlying geometry. Comparing these Kähler normal coordinates with the Riemannian normal coordinates defined via the exponential map we prove that their difference is a universal power series in the curvature tensor and its iterated covariant derivatives and devise an algorithm to calculate this power series to arbitrary order. As a byproduct we generalize Kähler normal coordinates to the class of complex affine manifolds with (1,1)-curvature tensor. Moreover we describe the Spencer connection on the infinite order Taylor series of the Kähler normal potential and obtain explicit formulas for the Taylor series of all relevant geometric objects on symmetric spaces.
The two-jet of the curvature tensor at some point of a pseudo-Riemannian manifold is called Einstein if the Ricci tensor is a multiple of the metric tensor at the given point and additionally its first two covariant derivatives vanish there. Following the Jet Isomorphism Theorem of pseudo-Riemannian geometry, we derive necessary and sufficient conditions for the Einstein property in terms of the symmetrization of the given two-jet (i.e. in terms of the Jacobi operator and its first two covariant derivatives along arbitrary geodesics emanating from the given point). A central role is played by the Weitzenböck formula for the Laplacian d delta + delta d acting on sections of the vector bundle of algebraic curvature tensors. As an application, we study linear Jacobi relations of order two on Einstein manifolds.
Recently examples of Riemannian homogeneous spaces with linear Jacobi relations were found. We calculate the Singer invariants of these spaces with the computer algebra program Maple and discuss the results by means of the Jet Isomorphism Theorem of pseudo-Riemannian geometry.
A classical theorem of Riemannian geometry, due in its original form to Cartan, states that the Taylor expansion of the metric in geodesic normal coordinates is a universal formal power series involving only the symmetrizations of the iterated covariant derivatives of the curvature tensor; this is known as the jet isomorphism theorem. In particular, it is in principle possible to reconstruct the jet of the curvature tensor from its symmetrization in geodesic normal coordinates, although this would certainly result in an unwieldy computation. In this paper we achieve the same goal by coordinate-free calculations, using only the intrinsic definition of the relevant Young symmetrizers.
We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kaehler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.
A submanifold of a Riemannian symmetric space is called parallel if its second fundamental form is a parallel section of the appropriate tensor bundle. We classify parallel submanifolds of the Grassmannian $\rmG^+_2(\R^{n+2})$ which parameterizes the oriented 2-planes of the Euclidean space $\R^{n+2}$\,. Our main result states that every complete parallel submanifold of $\rmG^+_2(\R^{n+2})$\,, which is not a curve, is contained in some totally geodesic submanifold as a symmetric submanifold. This result holds also if the ambient space is the non-compact dual of $\rmG^+_2(\R^{n+2})$\,.
We consider parallel submanifolds $M$ of a Riemannian symmetric space $N$ and study the question whether $M$ is extrinsically homogeneous in $N$\,, i.e.\ whether there exists a subgroup of the isometry group of $N$ which acts transitively on $M$\,. First, given a "2-jet" $(W,b)$ at some point $p\in N$ (i.e. $W\subset T_pN$ is a linear space and $b:W\times W\to W^\bot$ is a symmetric bilinear form)\,, we derive necessary and sufficient conditions for the existence of a parallel submanifold with extrinsically homogeneous tangent holonomy bundle which passes through $p$ and whose 2-jet at $p$ is given by $(W,b)$\,. Second, we focus our attention on complete, (intrinsically) {\em irreducible} parallel submanifolds of $N$\,. Provided that $N$ is of compact or non-compact type, we establish the extrinsic homogeneity of every complete, irreducible parallel submanifold of $N$ whose dimension is at least 3 and which is not contained in any flat of $N$\,.
Let $M$ and $N$ be Riemannian symmetric spaces and $f:M\to N$ be a parallel isometric immersion. We additionally assume that there exist simply connected, irreducible Riemannian symmetric spaces $M_i$ with $\dim(M_i)\geq 2$ for $i=1,...,r$ such that $M\cong M_1\times...\times M_r$ . As a starting point, we describe how the intrinsic product structure of $M$ is reflected by a distinguished, fiberwise orthogonal direct sum decomposition of the corresponding first normal bundle. Then we consider the (second) osculating bundle $\osc f$, which is a $\nabla^N$-parallel vector subbundle of the pullback bundle $f^*TN$, and establish the existence of $r$ distinguished, pairwise commuting, $\nabla^N$-parallel vector bundle involutions on $\osc f$ . Consequently, the "extrinsic holonomy Lie algebra" of $\osc f$ bears naturally the structure of a graded Lie algebra over the Abelian group which is given by the direct sum of $r$ copies of $\Z/2 \Z$ . Our main result is the following: Provided that $N$ is of compact or non-compact type, that $\dim(M_i)\geq 3$ for $i=1,...,r$ and that none of the product slices through one point of $M$ gets mapped into any flat of $N$, we can show that $f(M)$ is a homogeneous submanifold of $N$ .
We investigate parallel submanifolds of a Riemannian symmetric space $N$. The special case of a symmetric submanifold has been investigated by many authors before and is well understood. We observe that there is an intrinsic property of the second fundamental form which distinguishes full symmetric submanifolds from arbitrary full parallel submanifolds of $N$, usually called "1-fullness of $M$". Furthermore, for every parallel submanifold $M$ of $N$ we consider the pullback bundle $TN|M$ with its induced connection, which admits a distinguished parallel subbundle $osc M$, usually called the "second osculating bundle of $M$". If $M$ is a complete parallel submanifold of $N$, then we can describe the corresponding holonomy Lie algebra of $osc M$ by means of the second fundamental form of $M$ and the curvature tensor of $N$ at the origin. If moreover $N$ is simply connected and $M$ is even a full symmetric submanifold of $N$, then we will calculate the holonomy Lie algebra of $TN|M$ in an explicit form.
Due to its length and several inaccuracies, this article is no longer suggested for reading. Moreover, in the meantime certain results presented herein could be improved by the author in a non-trivial fashion. Instead, the reader is referred to the following two articles: The extrinsic holonomy Lie algebra of a parallel submanifold (arXiv:0904.2611), Extrinsic homogeneity of parallel submanifolds (arXiv:0904.2636).