Ann Glob Anal Geom https://doi.org/10.1007/s10455-018-9617-1 Virtual immersions and a characterization of symmetric spaces 1 2 Ricardo A. E. Mendes · Marco Radeschi Received: 14 March 2018 / Accepted: 22 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract We deﬁne virtual immersions, as a generalization of isometric immersions in a pseudo-Riemannian vector space. We show that virtual immersions possess a second funda- mental form, which is in general not symmetric. We prove that a manifold admits a virtual immersion with skew-symmetric second fundamental form, if and only if it is a symmetric space, and in this case the virtual immersion is essentially unique. Keywords Symmetric space · Isometric immersion · Pseudo-Euclidean space Mathematics Subject Classiﬁcation 49Q05 · 53A10 · 53C35 1 Introduction Often in Riemannian geometry, one needs to embed a Riemannian manifold into Euclidean or pseudo-Euclidean space. In this paper, we introduce a generalized and more “intrinsic” version of such embeddings and utilize them to give a new characterization of symmetric spaces. Given a Riemannian manifold M and an isometric immersion φ : M → V into a vector space (V , , ) endowed with a nondegenerate symmetric bilinear form (a pseudo-Euclidean vector
Annals of Global Analysis and Geometry – Springer Journals
Published: May 30, 2018
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