Abstract
The index of a Riemannian manifold is defined as the minimal codimension of a totally geodesic submanifold. In this note we discuss two recent results by the author and Olmos (Berndt and Olmos, On the index of symmetric spaces, preprint, arXiv:1401.3585) and some related topics. The first result states that the index of an irreducible Riemannian symmetric space is bounded from below by the rank of the symmetric space. The second result is the classification of all irreducible Riemannian symmetric spaces of noncompact type whose index is less or equal than three.
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Notes
- 1.
Note added in proof: Question 1 has now been answered by the author and Olmos in “Maximal totally geodesic submanifolds and index of symmetric spaces”, preprint arXiv:1405.0598. In the same paper the authors calculated the index of some further symmetric spaces and classified all irreducible Riemannian symmetric spaces M of noncompact type with i(M) ≤ 6.
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Berndt, J. (2014). Totally Geodesic Submanifolds of Riemannian Symmetric Spaces. In: Suh, Y.J., Berndt, J., Ohnita, Y., Kim, B.H., Lee, H. (eds) Real and Complex Submanifolds. Springer Proceedings in Mathematics & Statistics, vol 106. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55215-4_4
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