355 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
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
Pseudo-Riemannian Jacobi-Videv Manifolds
We exhibit several families of Jacobi-Videv pseudo-Riemannian manifolds which
are not Einstein. We also exhibit Jacobi-Videv algebraic curvature tensors
where the Ricci operator defines an almost complex structure
Curvature homogeneous spacelike Jordan Osserman pseudo-Riemannian manifolds
Let s be at least 2. We construct Ricci flat pseudo-Riemannian manifolds of
signature (2s,s) which are not locally homogeneous but whose curvature tensors
never the less exhibit a number of important symmetry properties. They are
curvature homogeneous; their curvature tensor is modeled on that of a local
symmetric space. They are spacelike Jordan Osserman with a Jacobi operator
which is nilpotent of order 3; they are not timelike Jordan Osserman. They are
k-spacelike higher order Jordan Osserman for ; they are k-timelike
higher order Jordan Osserman for , and they are not k timelike
higher order Jordan Osserman for .Comment: Update bibliography, fix minor misprint
Examples of signature (2,2) manifolds with commuting curvature operators
We exhibit Walker manifolds of signature (2,2) with various commutativity
properties for the Ricci operator, the skew-symmetric curvature operator, and
the Jacobi operator. If the Walker metric is a Riemannian extension of an
underlying affine structure A, these properties are related to the Ricci tensor
of A
Green functions of higher-order differential operators
The Green functions of the partial differential operators of even order
acting on smooth sections of a vector bundle over a Riemannian manifold are
investigated via the heat kernel methods. We study the resolvent of a special
class of higher-order operators formed by the products of second-order
operators of Laplace type defined with the help of a unique Riemannian metric
but with different bundle connections and potential terms. The asymptotic
expansion of the Green functions near the diagonal is studied in detail in any
dimension. As a by-product a simple criterion for the validity of the Huygens
principle is obtained. It is shown that all the singularities as well as the
non-analytic regular parts of the Green functions of such high-order operators
are expressed in terms of the usual heat kernel coefficients for a
special Laplace type second-order operator.Comment: 26 pages, LaTeX, 65 KB, no figures, some misprints and small mistakes
are fixed, final version to appear in J. Math. Phys. (May, 1998
Low-Energy Effective Action in Non-Perturbative Electrodynamics in Curved Spacetime
We study the heat kernel for the Laplace type partial differential operator
acting on smooth sections of a complex spin-tensor bundle over a generic
-dimensional Riemannian manifold. Assuming that the curvature of the U(1)
connection (that we call the electromagnetic field) is constant we compute the
first two coefficients of the non-perturbative asymptotic expansion of the heat
kernel which are of zero and the first order in Riemannian curvature and of
arbitrary order in the electromagnetic field. We apply these results to the
study of the effective action in non-perturbative electrodynamics in four
dimensions and derive a generalization of the Schwinger's result for the
creation of scalar and spinor particles in electromagnetic field induced by the
gravitational field. We discover a new infrared divergence in the imaginary
part of the effective action due to the gravitational corrections, which seems
to be a new physical effect.Comment: LaTeX, 42 page
The Heat Kernel Coefficients to the Matrix Schr\"odinger Operator
The heat kernel coefficients to the Schr\"odinger operator with a
matrix potential are investigated. We present algorithms and explicit
expressions for the Taylor coefficients of the . Special terms are
discussed, and for the one-dimensional case some improved algorithms are
derived.Comment: 16 pages, Plain TeX, 33 KB, no figure
Multiple reflection expansion and heat kernel coefficients
We propose the multiple reflection expansion as a tool for the calculation of
heat kernel coefficients. As an example, we give the coefficients for a sphere
as a finite sum over reflections, obtaining as a byproduct a relation between
the coefficients for Dirichlet and Neumann boundary conditions. Further, we
calculate the heat kernel coefficients for the most general matching conditions
on the surface of a sphere, including those cases corresponding to the presence
of delta and delta prime background potentials. In the latter case, the
multiple reflection expansion is shown to be non-convergent.Comment: 21 pages, corrected for some misprint
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