Connes duality in pseudo-Riemannian geometry

1 Pseudo - Riemannian Geometry

Generalized pseudo-Riemannian geometry

Generalized tensor analysis in the sense of Colombeau’s construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a “Fun...

Dimensional curvature identities on pseudo-Riemannian geometry

For a fixed n ∈ N, the curvature tensor of a pseudo-Riemannian metric, as well as its covariant derivatives, satisfy certain identities that hold on any manifold of dimension less or equal than n. In this paper, we re-elaborate recent results by Gilkey-Park-Sekigawa regarding these p-covariant curvature identities, for p = 0, 2. To this end, we use the classical theory of natural operations, th...

Pseudo-riemannian Geometry Calibrates Optimal Transportation

Given a transportation cost c : M M ! R, optimal maps minimize the total cost of moving masses from M to M . We …nd, explicitly, a pseudo-metric and a calibration form on M M such that the graph of an optimal map is a calibrated maximal submanifold. We de…ne the mass of space-like currents in spaces with inde…nite metrics.

Pseudo-riemannian Metrics in Models Based on Noncommutative Geometry

Several examples and models based on noncommutative differential calculi on commutative algebras indicate that a metric should be regarded as an element of the left-linear tensor product of the space of 1-forms with itself. We show how the metric compatibility condition with a linear connection generalizes to this framework.