In a Riemannian space, all tangent planes are nondegenerate and the sectional curvature is a continuous function. The planes of each tangent space form a compact set and the sectional curvature is bounded. In a pseudoriemannian space, the situation is quite different. The sectional curvature is only defined on nondegenerate planes, and those form a noncompact set whenever the dimension is at least 3. Our approach differs from previous studies in that we begin by expressing the sectional curvature at a point of a three-dimensional Lorentzian manifold as a rational function which is a ratio of quadrics. In dimension three, the sectional curvature must become unbounded near all null planes with at most four exceptions whenever it is not constant at a point. In higher dimensions, the degenerate planes which are indeterminate lie in a set of codimension at least 3. The set of spacelike directions which determine pencils of planes with unbounded sectional curvature form an open dense subset of the set of all spacelike directions, with the complement of codimension at least 2.
We discuss conditions under which a lens space is sth order flat.
We consider particular types of discrete approximations to tensor fields on manifolds suggested by triangulations. The approximations are objects of finite geometrical extent, parameterized by a finite set of numbers, so they are suitable for numerical computations. We study the limiting behaviour of sequences of approximations and construct the theory so that the limits are tensor fields on the manifold. We propose a Cauchy criterion for our approximations which guarantees convergence to a limit. The specific examples include geodesic approximations to Riemannian and pseudoriemannian manifolds.
The space of principal connections of principal circle bundles over $T^2$ and over $S^2$ is characterized. In particular, this latter includes the famous Hopf fibration.
This is a new geometric study in the theory of partial differential equations: an application of Fourier integral operators to equations with distributional coefficients. As an example, a derivation of the existence and spectrum of Hawking's radiation from a collapsing star is given.
We find all metric tensors on the total space of a fiber bundle which agree vertically with a given metric tensor on the model fiber and horizontally with one on the base space. We give some examples, and an application to Polyakov strings.
We give a complete classification in canonical forms on finite-dimensional vector spaces over the real numbers.
I characterize the existence of horizontal path lifts for general connections with a new property that also gives fresh insight into linear and G-connections.
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