Restrictions on submanifolds via focal radius bounds
We give an optimal estimate for the norm of any submanifold’s second fundamental form in terms of its focal radius and the lower sectional curvature bound of the ambient manifold. This is a special case of a similar theorem for intermediate Ricci curvature, and leads to a C1,α compactness result for...
| Autores: | , |
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| Formato: | artículo |
| Fecha de publicación: | 2020 |
| País: | España |
| Recursos: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/741620 |
| Acesso em linha: | https://hdl.handle.net/10486/741620 https://dx.doi.org/10.4310/MRL.2020.v27.n1.a7 |
| Access Level: | acceso abierto |
| Palavra-chave: | Focal Radius rigidity projective space positive curvature Matemáticas |
| Resumo: | We give an optimal estimate for the norm of any submanifold’s second fundamental form in terms of its focal radius and the lower sectional curvature bound of the ambient manifold. This is a special case of a similar theorem for intermediate Ricci curvature, and leads to a C1,α compactness result for submanifolds, as well as a “soul-type” structure theorem for manifolds with nonnegative kth–intermediate Ricci curvature that have a closed submanifold with dimension ≥k and infinite focal radius. To prove these results, we use a new comparison lemma for Jacobi fields from [18] that exploits Wilking’s transverse Jacobi equation. The new comparison lemma also yields new information about group actions, Riemannian submersions, and submetries, including generalizations to intermediate Ricci curvature of results of Chen and Grove. None of these results can be obtained with just classical Riccati comparison (see Subsection 3.1 for details.) |
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