Riemannian Geometries Without The "Riemann" Part

Discussion in 'Math Research' started by Rock Brentwood, Sep 15, 2011.

  1. 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

      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
    (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

    But I know of no generalization of Riemannian geometry that deals with
    these issues.
    Rock Brentwood, Sep 15, 2011
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