P. Gilkey

Publications68
h-index11
Citations440
Highly Influential Citations25
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The Geometry of Walker Manifolds
TLDR
In this Chapter, we introduce the algebraic structures that we will be using; the reader may want to read the next Chapter (which deals with the corresponding geometric structures) before reading this Chapter. Expand
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Universal curvature identities
Abstract 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… Expand
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Applications of Affine and Weyl Geometry
TLDR
Pseudo-Riemannian geometry is, to a large extent, the study of the Levi-Civita connection, which is the unique torsion-free connection compatible with the metric structure. Expand
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Algebraic curvature tensors whose skew-symmetric curvature operator has constant rank 2
TLDR
We classify the algebraic curvature tensors so R(ċ) has constant rank 2 and show these are geometrically realizable by hypersurfaces in flat spaces. Expand
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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… Expand
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GEOMETRIC REALIZATIONS OF KAEHLER AND OF PARA-KAEHLER CURVATURE MODELS
We show that every Kaehler algebraic curvature tensor is geometrically realizable by a Kaehler manifold of constant scalar curvature. We also show that every para-Kaehler algebraic curvature tensor… Expand
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Geometric realizations of Hermitian curvature models
We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact… Expand
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Curvature homogeneous signature (2,2) manifolds
We exhibit a family of generalized plane wave manifolds of signature (2,2). The geodesics in these manifolds extend for infinite time (i.e. they are complete), they are spacelike and timelike Jordan… Expand
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Conformally Osserman manifolds and self-duality in Riemannian geometry
We study the spectral geometry of the conformal Jacobi operator on a 4-dimensional Riemannian manifold (M,g). We show that (M,g) is conformally Osserman if and only if (M,g) is self-dual or anti… Expand
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Geometric Realizations of Curvature
Introduction and Statement of Results Representation Theory Connections, Curvature, and Differential Geometry Affine Geometry Kahler and Para-Kahler Geometry Affine Geometry Riemannian Geometry… Expand
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