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2011

Cartan's Structural Equations for Singular Manifolds

Stoica, Ovidiu Cristinel

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The classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames.

In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent bundle with a symmetric bilinear form which is allowed to become degenerate.
But if the fundamental tensor is allowed to be degenerate, there are some obstructions in constructing the geometric objects normally associated to the fundamental tensor.
Also, local orthonormal frames and coframes no longer exist, as well as the metric connection and its curvature operator.

Cartan's Structural Equations for Singular Manifolds · Around

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Sur les géodésiques des espaces de Riemann singuliers
G. C. Moisil · 1940
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Sur les invariants des espaces de Riemann singuliers
G. Vrănceanu · 1942
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Semi-Riemannian Geometry with Applications to Relativity
B. O’Neill · 1983
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Degenerate manifolds
D. Kupeli · 1987
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The Geometry of Kerr Black Holes
B. O’Neill · 1995
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Singular Semi-Riemannian Geometry
D. Kupeli · 1996
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