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Topic: Riemannian Geometries Without The "Riemann" Part
Replies: 2   Last Post: Sep 17, 2011 10:30 AM

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Rock Brentwood

Posts: 121
Registered: 6/18/10
Riemannian Geometries Without The "Riemann" Part
Posted: Sep 15, 2011 7:49 AM
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A Riemannian geometry is an manifold equipped with an affine
connection and a positive definite metric (or negative definite). This
generalizes to pseudo-Riemannian geometries where the metric continues
to be of a definite signature and non-degenerate, 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 well-defined 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 non-zero
"dimension")

  del^2 = d_x^2 + d_y^2 + d_z^2 -- contravariant metric (degenerate
with 3 non-zero "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 non-uniqueness 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.



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