Searcharxiv⌕ Search

arXiv subjects

Peter B. Gilkey

Publications and source records attributed to Peter B. Gilkey.

14 recordsLinked to original sources
arXiv2020

Derived heat trace asymptotics for the de Rham and Dolbeault complexes

We examine the derived heat trace asymptotics in both the real and the complex settings for a generalized Witten perturbation. If the dimension is even, in the real context we show the integral of the local density for the derived heat trace asymptotics is half the Euler characteristic of the underlying manifold. In the complex context, we assume the underlying geometry is Kähler and show the integral of the local density for the derived heat trace asymptotics defined by the Dolbeault complex is a characteristic number of the complex tangent bundle and the twisting vector bundle. We identify this characteristic number if the real dimension is $2$ or $4$. In both the real and complex settings, the local density differs from the corresponding characteristic class by a divergence term.

math.DG↗
arXiv2016

Geodesic completeness for Type~A surfaces

Type A surfaces are the locally homogeneous affine surfaces which can be locally described by constant Christoffel symbols. We address the issue of the geodesic completeness of these surfaces: we show that some models for Type A surfaces are geodesically complete, that some others admit an incomplete geodesic but model geodesically complete surfaces, and that there are also others which do not model any complete surface. Our main result provides a way of determining whether a given set of constant Christoffel symbols can model a complete surface.

math.DG↗
arXiv2011

Universal curvature identities II

We show that any universal curvature identity which holds in the Riemannian setting extends naturally to the pseudo-Riemannian setting. Thus the Euh-Park-Sekigawa identity also holds for pseudo-Riemannian manifolds. We study the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that as in the Riemannian setting, they are given solely in terms of curvature (and not in terms of covariant derivatives of curvature) even in the pseudo-Riemannian setting.

math.DG↗
arXiv2008

Heat Content, Heat Trace, and Isospectrality

We study the heat content function, the heat trace function, and questions of isospectrality for the Laplacian with Dirichlet boundary conditions on a compact manifold with smooth boundary in the context of finite coverings and warped products.

math.AP↗
arXiv2008

Geometrical representations of equiaffine curvature operators

We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent bundle of some smooth manifold.

math.DG↗
arXiv2002

Jordan Szabo algebraic covariant derivative curvature tensors

We show that if $\nabla R$ is a Jordan Szabo algebraic covariant derivative curvature tensor on a vector space of signature (p,q), where q is odd and p is less than q or if q is congruent to 2 mod 4 and if p is less than q-1, then $\nabla R=0$. This algebraic result yields an elementary proof of the geometrical fact that any pointwise totally isotropic pseudo-Riemannian manifold with such a signature (p,q) is locally symmetric.

math.DG↗
arXiv2002

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.

math.DG↗