Annals of mathematics studies

* Basic Algebraic Notions* Basic Geometrical Notions* Walker Structures* Three-Dimensional Lorentzian Walker Manifolds* Four-Dimensional Walker Manifolds* The Spectral Geometry of the Curvature Tensor* Hermitian Geometry* Special Walker Manifolds

The Geometry of Walker Manifolds

Suppose that we havethe unit Euclidean ball in Rn and construct new bodies using three op- erations — linear transformations, closure in the radial metric, and multiplicative summation defined by kxkK+0L = p kxkKkxkL. We prove that in dimension 3 this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in L0 that naturally extends the corresponding properties of Lp-spaces with p 6 0, and show that the procedure described above gives exactly the unit balls of subspaces of L0 in everydimension. We provideFourier analytic and geometric characterizations of spaces embedding in L0, and prove several facts confirming the place of L0 in the scale of Lp-spaces.

The geometry of $L_0$

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and gives necessary and sufficient conditions for a modified Riemannian extension to be Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3, 3), whose Jacobi operators have non-trivial Jordan normal form and which are not nilpotent. We present new four-dimensional results in Osserman geometry.

The geometry of modified Riemannian extensions

It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson–Walker warped product, if the gradient of the potential function is nonnull, and to a plane wave, if the gradient of the potential function is null. The latter gradient Ricci solitons are necessarily steady.

Locally Conformally Flat Lorentzian Gradient Ricci Solitons

We determine the local structure of all pseudo-Riemannian manifolds of dimensions greater than 3 whose Weyl conformal tensor is parallel and has rank 1 when treated as an operator acting on exterior 2-forms at each point. If one fixes three discrete parameters: the dimension, the metric signature (with at least two minuses and at least two pluses), and a sign factor accounting for semidefiniteness of the Weyl tensor, then the local-isometry types of our metrics correspond bijectively to equivalence classes of surfaces with equiaffine projectively flat torsionfree connections; the latter equivalence relation is provided by unimodular affine local diffeomorphisms. The surface just mentioned arises, locally, as the leaf space of a codimension-two parallel distribution on the pseudo-Riemannian manifold in question, naturally associated with its metric. We construct examples showing that the leaves of this distribution may form a fibration with the base which is a closed surface of any prescribed diffeomorphic type. Our result also completes a local classification of pseudo-Riemannian metrics with parallel Weyl tensor that are neither conformally flat nor locally symmetric: for those among such metrics which are not Ricci-recurrent, the Weyl tensor has rank 1, and so they belong to the class discussed in the previous paragraph; on the other hand, the Ricci-recurrent ones have already been classified by the second author.

/pdf/projectively-flat-surfaces-null-parallel-distributions-and-5cweahialf.pdf

Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds

This is a final step in a local classification of pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric.

/pdf/the-local-structure-of-conformally-symmetric-manifolds-2mr6pf3rcd.pdf

The local structure of conformally symmetric manifolds

https://people.math.osu.edu/derdzinski.1/preprints/jgp08.pdf

On compact manifolds admitting indefinite metrics with parallel Weyl tensor

We provide a coordinate-free version of the local classification, due to Walker [Q. J. Math. 1, 69 (1950)], of null parallel distributions on pseudo-Riemannian manifolds. The underlying manifold is realized, locally, as the total space of a fiber bundle, each fiber of which is an affine principal bundle over a pseudo-Riemannian manifold. All structures just named are naturally determined by the distribution and the metric, in contrast with the noncanonical choice of coordinates in the usual formulation of Walker’s theorem.

/pdf/walker-s-theorem-without-coordinates-2vtk1caw0c.pdf

Walker’s theorem without coordinates

It is shown that in every dimension n=3j+2, j=1,2,3,..., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite metric signatures. The manifolds in question are diffeomorphic to nontrivial torus bundles over the circle. They all arise from a construction that a priori yields bundles over the circle, having as the fibre either a torus, or a 2-step nilmanifold with a complete flat torsionfree connection; our argument only realizes the torus case.

Witold Roter

Papers

Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds

The local structure of conformally symmetric manifolds

On compact manifolds admitting indefinite metrics with parallel Weyl tensor

Walker’s theorem without coordinates

Compact pseudo-Riemannian manifolds with parallel Weyl tensor