Abstract.
In this paper, we are interested in extending the study of spherical curves in R 3 to the submanifolds in the Euclidean space R n+p. More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R n+p lies on the hypersphere S n+p−1(c) (standardly imbedded sphere in R n+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S n(c) (cf. Theorem 4.1).
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Alodan, H., Deshmukh, S. Spherical submanifolds in a Euclidean space. Mh Math 152, 1–11 (2007). https://doi.org/10.1007/s00605-007-0453-9
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DOI: https://doi.org/10.1007/s00605-007-0453-9