Geodesic
Almost-Riemannian Structures on Banach Manifolds: The Morse Lemma and the Darboux Theorem
Canadian journal of mathematics, Tome 28 (1976) no. 3, pp. 640-652

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper we introduce the notion of an almost Riemannian manifold. Briefly speaking, an almost-Riemannian structure on a Banach manifold is a generalization of the notion of a Riemannian structure on a Hilbert manifold, common examples would be manifolds of maps modelled on the Sobolev spaces Lkp. The successful use of weak Riemannian structures in hard problems has been given by Ebin [2] and Ebin and Marsden [10]. 
Tromba, A. J. Almost-Riemannian Structures on Banach Manifolds: The Morse Lemma and the Darboux Theorem. Canadian journal of mathematics, Tome 28 (1976) no. 3, pp. 640-652. doi: 10.4153/CJM-1976-064-x
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