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P. Gilkey

Publications and source records attributed to P. Gilkey.

At least 19 recordsLinked to original sources
arXiv2019

On Distinguished local coordinates for locally homogeneous affine surfaces

We give a new short self-contained proof of the result of Opozda [B. Opozda, A classification of locally homogeneous connections on 2-dimensional manifolds, Differential Geom. Appl. 21 (2004), 173-198.] classifying the locally homogeneous torsion free affine surfaces and the extension to the case of surfaces with torsion due to Arias-Marco and Kowalski [T. Arias-Marco and O. Kowalski, Classification of locally homogeneous linear connections with arbitrary torsion on 2-dimensional manifolds, Monatsh. Math. 153 (2008), 1-18.]. Our approach rests on a direct analysis of the affine Killing equations and is quite different than the approaches taken previously in the literature.

math.DG↗
arXiv2018

Applications of PDEs to the study of affine surface geometry

If $\mathcal{M}=(M,\nabla)$ is an affine surface, let $\mathcal{Q}(\mathcal{M}):=\ker(\mathcal{H}+\frac1{m-1}\rho_s)$ be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let $\tilde{\mathcal{M}}=(M,\tilde\nabla)$ be another affine structure on $M$ which is strongly projectively flat. We show that $\mathcal{Q}(\mathcal{M})=\mathcal{Q}(\tilde{\mathcal{M}})$ if and only if $\nabla=\tilde\nabla$ and that $\mathcal{Q}(\mathcal{M})$ is linearly equivalent to $\mathcal{Q}(\tilde{\mathcal{M}})$ if and only if $\mathcal{M}$ is linearly equivalent to $\tilde{\mathcal{M}}$. We use these observations to classify the flat Type~$\mathcal{A}$ connections up to linear equivalence, to classify the Type~$\mathcal{A}$ connections where the Ricci tensor has rank 1 up to linear equivalence, and to study the moduli spaces of Type~$\mathcal{A}$ connections where the Ricci tensor is non-degenerate up to affine equivalence.

math.DG↗
arXiv2018

Solutions to the affine quasi-Einstein equation for homogeneous surfaces

We examine the space of solutions to the affine quasi--Einstein equation in the context of homogeneous surfaces. As these spaces can be used to create gradient Yamabe solitions, conformally Einstein metrics, and warped product Einstein manifolds using the modified Riemannian extension, we provide very explicit descriptions of these solution spaces. We use the dimension of the space of affine Killing vector fields to structure our discussion as this provides a convenient organizational framework.

math.DG↗
arXiv2018

Affine surfaces which are K\"ahler, para-K\"ahler, or nilpotent K\"ahler

Motivated by the construction of Bach flat neutral signature Riemannian extensions, we study the space of parallel trace free tensors of type $(1,1)$ on an affine surface. It is shown that the existence of such a parallel tensor field is characterized by the recurrence of the symmetric part of the Ricci tensor.

math.DG↗
arXiv2017

The geometry of locally symmetric affine surfaces

We examine the local geometry of affine surfaces which are locally symmetric. There are 6 non-isomorphic local geometries. We realize these examples as Type A, Type B, and Type C geometries using a result of Opozda and classify the relevant geometries up to linear isomorphism. We examine the geodesic structures in this context. Particular attention is paid to the Lorentzian analogue of the hyperbolic plane and to the pseudosphere.

math.DG↗
arXiv2016

Homogeneous affine surfaces: Moduli spaces

We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type $\mathcal{A}$ surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type $\mathcal{B}$ surfaces which are not Type $\mathcal{A}$ we show the corresponding moduli space is a simply connected real analytic 4-dimensional manifold with second Betti number equal to $1$.

math.DG↗
arXiv2015

Homogeneous affine surfaces: affine Killing vector fields and Gradient Ricci solitons

The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of these other two families, we examine the Lie algebra of affine Killing vector fields and we give a complete classification of the homogeneous affine gradient Ricci solitons. The rank of the Ricci tensor plays a central role in our analysis.

math.DG↗
arXiv2015

Anti-invariant Riemannian Submersions

We give a general Lie-theoretic construction for anti-invariant almost Hermitian Riemannian submersions, anti-invariant quaternion Riemannian submersions, anti-invariant para-Hermitian Riemannian submersions, anti-invariant para-quaternion Riemannian submersions, and anti-invariant octonian Riemannian submersions. This yields many compact Einstein examples.

math.DG↗
arXiv2014

Real analytic complete non-compact surfaces in Euclidean space with finite total curvature arising as solutions to ODEs

We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associated roots of the ODEs are real and distinct, we give a universal upper bound for the total Gauss curvature of the surface which depends only on the orders of the ODEs and we show that the total Gauss curvature of the surface vanishes if the ODEs are second order. We examine when the surfaces are asymptotically minimal.

math.DG↗
arXiv2014

Homothety Curvature Homogeneity and Homothety Homogeneity

We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanzurova are genuine generalizations of the ordinary notion of k-curvature homogeneity. The homothety group plays an essential role in the analysis. We give a complete classification of homothety homogeneous manifolds which are not homogeneous and which are not VSI and show that such manifolds are cohomogeneity one. We also give a complete description of the local geometry if the homothety character defines a split extension.

math.DG↗
arXiv2014

Heat content with singular initial temperature and singular specific heat

Let (M,g) be a compact Riemannian manifold without boundary. Let D be a compact subdomain of M with smooth boundary. We examine the heat content asymptotics for the heat flow from D into M where both the initial temperature and the specific heat are permitted to have controlled singularities on the boundary of D. The operator driving the heat process is assumed to be an operator of Laplace type

math.AP↗
arXiv2013

Homothety Curvature Homogeneity

We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Van\v{z}urov\'a are genuine generalizations of the ordinary notion of $k$-curvature homogeneity. The homothety group plays an essential role in the analysis.

math.DG↗
arXiv2013

Heat flow out of a compact manifold

We discuss the heat content asymptotics associated with the heat flow out of a smooth compact manifold in a larger compact Riemannian manifold. Although there are no boundary conditions, the corresponding heat content asymptotics involve terms localized on the boundary. The classical pseudo-differential calculus is used to establish the existence of the complete asymptotic series and methods of invariance theory are used to determine the first few terms in the asymptotic series in terms of geometric data. The operator driving the heat process is assumed to be an operator of Laplace type.

math.AP↗