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Riemann extensions and affine differential geometry. (English) Zbl 0647.53008
It was shown by A. G. Walker [Convegno Internaz. Geometria Differenz., Venice/Italy 1953, 64-70 (1954; Zbl 0056.154)] that a torsion-free affine connection on a manifold canonically determines a pseudo-Riemannian metric on the cotangent bundle, called the Riemann- extension of the affine connection. In this paper the author gives an intrinsic definition of the Riemann-extension and shows that by making use of this pseudo-Riemannian metric it is possible to define an affine immersion of manifolds in affine differential geometry without making a suitable choice of normal planes.
Reviewer: Li An-Min
MSC:
53A15Affine differential geometry
53B05Linear and affine connections
53B25Local submanifolds
References:
[1]Walker, A.G., ’Riemann extensions for non-Riemannian spaces’, Convegno di Geometria Differenziale Venezia (1953), pp. 64–70.
[2]Walker, A.G., ’A canonical form for a Riemannian space with a parallel field of null planes’, Quart. Journ. Math. (Oxford) (2), 1 (1950), pp.64–79.