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Riemannian Geometries Without The "Riemann" Part
Posted:
Sep 15, 2011 7:49 AM


A Riemannian geometry is an manifold equipped with an affine connection and a positive definite metric (or negative definite). This generalizes to pseudoRiemannian geometries where the metric continues to be of a definite signature and nondegenerate, but not necessarily positive (or negative) definite.
Consider, however, the following geometry given by the following metric written as a line element: ds^2 = dx^2 + dy^2 + dz^2  (1/t) dt^2 over BOTH the domains t < 0 and t > 0. Over t < 0 it is Riemannian, over t > 0 it is Lorentzian. But there is no overarching mathematical theory I know of which describes the entire geometry  including the part where t = 0.
Nonetheless, it has welldefined geodesics (except on the t = 0 surface, itself).
Consider, as a second example, the geometry underlying non relativistic spacetime. Here, one has mutually independent covariant and contravariant metrics, represented by the following invariants ds^2 = dt^2  covariant metric (degenerate with 1 nonzero "dimension")
del^2 = d_x^2 + d_y^2 + d_z^2  contravariant metric (degenerate with 3 nonzero "dimensions")
Instead of multiplying out to the unit matrix, the matrix representations of the two metrics multiply out to 0.
This is an affine geometry, when equipped with an affine connection (but this gets to another problem: the nonuniqueness of a connection canonically associated with the covariant and contravariant metrics).
Both of these cases point to a generalization of (pseudo)Riemannian geometry in which (a) the covariant and contravariant metric need not be inversely related (b) the signature of the metric(s) may change  thus leading to the concepts of (c) a "signature domain" (d) a "signature domain boundary"
In such generalized geometries, one may proceed to ask what the associated (generalized) orthogonal group for a given covariant/ contravariant metric signature combination. Then the question arises as to how this changes on a signature boundary  an issue that may link up with the question of the (a) real forms of the complex orthogonal group and (b) the contractions of the real orthogonal group.
But I know of no generalization of Riemannian geometry that deals with these issues.



