Abstract
E. Binz [1] considered two canonical Riemannian metrics on the space of embeddings of a closed (n−1) dimensional manifold into ℝn, and computed the geodesic sprays. Here we consider the space of immersions Imm (M, N) whereM is without boundary, and we compute the covariant derivative (in the form of its connector) and the Riemannian curvature of one of these metrics, the non trivial one. The setting is close to that used byP. Michor [2], and we refer the reader to this paper for notation.
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References
Binz, E.: Two natural metrics and their covariant derivatives on a manifold of embeddings. Mh. Math.89, 275–288 (1980).
Michor, P.: Manifolds of Differentiable Mappings. Math. Ser. 3. Orpington: Shiva. 1980.
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Kainz, G. A metric on the manifold of immersions and its Riemannian curvature. Monatshefte für Mathematik 98, 211–217 (1984). https://doi.org/10.1007/BF01507749
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DOI: https://doi.org/10.1007/BF01507749