Riemann extensions and affine differential geometry |
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Authors: | Tom Willmore |
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Institution: | 1. Department of Mathematics, University of Durham, Durham, DHI 3LE, England
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Abstract: | In order to define an affine immersion of manifolds in affine differential geometry, it is necessary to choose a set of normal planes to the immersed manifold. The theory is then developed after this choice has been made. However, it was shown by A.G.Walker WA1] that a torsion-free affine connexion on a manifold determines canonically a pseudo-Riemannian metric on the cotangent bundle, called the Riemann-extension of the affine connexion. By making use of this pseudo-Riemannian metric it is possible to define an affine immersion without making a suitable choice of normal planes. |
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