Searcharxiv⌕ Search

arXiv subjects

P. Gilkey

Publications and source records attributed to P. Gilkey.

At least 37 records · Page 2Linked to original sources
arXiv2013

Transplanting geometrical structures

We say that a germ G of a geometric structure can be transplanted into a manifold M if there is a suitable geometric structure on M which agrees with G on a neighborhood of some point P of M. We show for a wide variety of geometric structures that this transplantation is always possible provided that M does in fact admit some such structure of this type. We use this result to show that a curvature identity which holds in the category of compact manifolds admitting such a structure holds for germs as well and we present examples illustrating this result. We also use this result to show geometrical realization problems which can be solved for germs of structures can in fact be solved in the compact setting as well.

math.DG↗
arXiv2012

Neumann heat content asymptotics with singular initial temperature and singular specific heat

We study the asymptotic behavior of the heat content on a compact Riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions imposing Robin boundary conditions. Assuming the existence of a complete asymptotic series we determine the first three terms in that series. In addition to the general setting, the interval is studied in detail as are recursion relations among the coefficients and the relationship between the Dirichlet and Robin settings.

math.AP↗
arXiv2012

Homogeneous 4-dimensional Kaehler--Weyl Structures

Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kaehler-Weyl structure; this structure is locally conformally Kaehler if and only if the alternating Ricci tensor vanishes. The alternating Ricci tensor takes values in a certain representation space. In this paper, we show that any algebraic possibility in this representation space can in fact be geometrically realized by a left-invariant Kaehler-Weyl structure on a 4-dimensional Lie group in either the Hermitian or the para-Hermitian setting.

math.DG↗
arXiv2012

Homogeneity of Lorentzian three-manifolds with recurrent curvature

k-Curvature homogeneous three-dimensional Walker metrics are described for k=0,1,2. This allows a complete description of locally homogeneous three-dimensional Walker metrics, showing that there exist exactly three isometry classes of such manifolds. As an application one obtains a complete description of all locally homogeneous Lorentzian manifolds with recurrent curvature. Moreover, potential functions are constructed in all the locally homogeneous manifolds resulting in steady gradient Ricci and Cotton solitons.

math.DG↗
arXiv2012

Universal curvature identities III

We examine universal curvature identities for pseudo-Riemannian manifolds with boundary. We determine the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that they are given solely in terms of curvature {and the second fundamental form and do not involve covariant derivatives thus generalizing a conjecture of Berger to this context.

math.DG↗
arXiv2012

(para)-Kähler Weyl structures

We work in both the complex and in the para-complex categories and examine (para)-Kähler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kaehler Weyl algebraic curvature tensor is in fact Riemannian in dimension at least 6; this yields as a geometric consequence that any (para)-Kaehler Weyl geometric structure is trivial if the dimension is at least 6. By contrast, the 4 dimensional setting is, as always, rather special as it turns out that there are (para)-Kaehler Weyl algebraic curvature tensors which are not Riemannian in dimension 4. Since every (para)-Kaehler Weyl algebraic curvature tensor is geometrically realizable and since every 4 dimensional Hermitian manifold admits a unique (para)-Kaehler Weyl structure, there are also non-trivial 4 dimensional Hermitian (para)-Kaehler Weyl manifolds.

math.DG↗
arXiv2012

Heat trace asymptotics and the Gauss-Bonnet Theorem for general connections

We examine the local super trace asymptotics for the de Rham complex defined by an arbitrary super connection on the exterior algebra. We show, in contrast to the situation in which the connection in question is the Levi-Civita connection, that these invariants are generically non-zero in positive degree and that the critical term is not the Pfaffian.

math.AP↗
arXiv2012

Heat content asymptotics with singular data

We study the asymptotic behaviour of the heat content on a compact Riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions. Assuming the existence of a complete asymptotic series we determine the first three terms in that series. In addition to the general setting, the interval is studied in detail.

math.AP↗
arXiv2011

Growth of heat trace coefficients for locally symmetric spaces

We study the asymptotic behavior of the heat trace coefficients $a_n$ as n tends to infinity for the scalar Laplacian in the context of locally symmetric spaces. We show that if the Plancherel measure of a noncompact type symmetric space is polynomial, then these coefficients decay like 1/n!. On the other hand, for even dimensional locally rank 1-symmetric spaces, one has $|a_n|$ grows like C^n* n! for some C>0; we conjecture this is the case in general if the associated Plancherel measure is not polynomial. These examples show that growth estimates conjectured by Berry and Howls are sharp. We also construct examples of locally symmetric spaces which are not irreducible, which are not flat, and so that only a finite number of the heat trace coefficients are non-zero.

math.AP↗
arXiv2011

Kähler-Weyl manifolds of dimension 4

We determine the space of algebraic pseudo-Hermitian Kähler-Weyl curvature tensors and the space of para-Hermitian Kähler-Weyl curvature tensors in dimension 4 and show that every algebraic possibility is geometrically realizable. We establish the Gray identity for pseudo-Hermitian Weyl manifolds and for para-Hermitian Weyl manifolds in arbitrary dimension.

math.DG↗
arXiv2011

Hardy inequality and heat semigroup estimates for Riemannian manifolds with singular data

Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on bd(D), and non-negative initial condition. We show that these upper bounds are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting in $L^2(D)$ satisfies a strong Hardy inequality with weight $r^2$, (ii) the initial temperature distribution, and the specific heat of D are given by $r^{-a}$ and $r^{-b}$ respectively, where $r$ is the distance to the boundary, and 1<a<2, 1<b<2.

math.SP↗
arXiv2011

Universal curvature identities

We study scalar and symmetric 2-form valued universal curvature identities. We use this to establish the Gauss-Bonnet theorem using heat equation methods, to give a new proof of a result of Kuz'mina and Labbi concerning the Euler-Lagrange equations of the Gauss-Bonnet integral, and to give a new derivation of the Euh-Park-Sekigawa identity.

math.DG↗
arXiv2010

Higher-dimensional Osserman metrics with non-nilpotent Jacobi operators

We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n,n) for any n which is at least 3. These examples admit a natural almost para-Hermitian structure and are semi para-complex Osserman with non-trivial Jordan normal form as well; they neither satisfy the third Gray identity nor are they integrable.

math.DG↗
arXiv2009

The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group cyclic of odd prime order p. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p=n=3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.

math.DG↗