413 research outputs found
Higher order Jordan Osserman Pseudo-Riemannian manifolds
We study the higher order Jacobi operator in pseudo-Riemannian geometry. We
exhibit a family of manifolds so that this operator has constant Jordan normal
form on the Grassmannian of subspaces of signature (r,s) for certain values of
(r,s). These pseudo-Riemannian manifolds are new and non-trivial examples of
higher order Osserman manifolds
Covariant techniques for computation of the heat kernel
The heat kernel associated with an elliptic second-order partial differential
operator of Laplace type acting on smooth sections of a vector bundle over a
Riemannian manifold, is studied. A general manifestly covariant method for
computation of the coefficients of the heat kernel asymptotic expansion is
developed. The technique enables one to compute explicitly the diagonal values
of the heat kernel coefficients, so called Hadamard-Minackshisundaram-De
Witt-Seeley coefficients, as well as their derivatives. The elaborated
technique is applicable for a manifold of arbitrary dimension and for a generic
Riemannian metric of arbitrary signature. It is very algorithmic, and well
suited to automated computation. The fourth heat kernel coefficient is computed
explicitly for the first time. The general structure of the heat kernel
coefficients is investigated in detail. On the one hand, the leading derivative
terms in all heat kernel coefficients are computed. On the other hand, the
generating functions in closed covariant form for the covariantly constant
terms and some low-derivative terms in the heat kernel coefficients are
constructed by means of purely algebraic methods. This gives, in particular,
the whole sequence of heat kernel coefficients for an arbitrary locally
symmetric space.Comment: 31 pages, LaTeX, no figures, Invited Lecture at the University of
Iowa, Iowa City, April, 199
Complete curvature homogeneous pseudo-Riemannian manifolds
We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian
manifolds which are modeled on irreducible symmetric spaces and which are not
locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some
of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.Comment: Update paper to fix misprints in original versio
Covariant Algebraic Method for Calculation of the Low-Energy Heat Kernel
Using our recently proposed covariant algebraic approach the heat kernel for
a Laplace-like differential operator in low-energy approximation is studied.
Neglecting all the covariant derivatives of the gauge field strength
(Yang-Mills curvature) and the covariant derivatives of the potential term of
third order and higher a closed formula for the heat kernel as well as its
diagonal is obtained. Explicit formulas for the coefficients of the asymptotic
expansion of the heat kernel diagonal in terms of the Yang-Mills curvature, the
potential term and its first two covariant derivatives are obtained.Comment: 19 pages, Plain TeX, 44 KB, no figure
Spectral geometry, homogeneous spaces, and differential forms with finite Fourier series
Let G be a compact Lie group acting transitively on Riemannian manifolds M
and N. Let p be a G equivariant Riemannian submersion from M to N. We show that
a smooth differential form on N has finite Fourier series if and only if the
pull back has finite Fourier series on
Graphical Classification of Global SO(n) Invariants and Independent General Invariants
This paper treats some basic points in general relativity and in its
perturbative analysis. Firstly a systematic classification of global SO(n)
invariants, which appear in the weak-field expansion of n-dimensional
gravitational theories, is presented. Through the analysis, we explain the
following points: a) a graphical representation is introduced to express
invariants clearly; b) every graph of invariants is specified by a set of
indices; c) a number, called weight, is assigned to each invariant. It
expresses the symmetry with respect to the suffix-permutation within an
invariant. Interesting relations among the weights of invariants are given.
Those relations show the consistency and the completeness of the present
classification; d) some reduction procedures are introduced in graphs for the
purpose of classifying them. Secondly the above result is applied to the proof
of the independence of general invariants with the mass-dimension for the
general geometry in a general space dimension. We take a graphical
representation for general invariants too. Finally all relations depending on
each space-dimension are systematically obtained for 2, 4 and 6 dimensions.Comment: LaTex, epsf, 60 pages, many figure
The structure of algebraic covariant derivative curvature tensors
We use the Nash embedding theorem to construct generators for the space of
algebraic covariant derivative curvature tensors
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