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Riemannian Questions with a Fundamental Differential System

  • Rui Albuquerque
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)

Abstract

We introduce the reader to a fundamental exterior differential system of Riemannian geometry which arises naturally with every oriented Riemannian n + 1-manifold M. Such system is related to the well-known metric almost contact structure on the unit tangent sphere bundle SM, so we endeavor to include the theory in the field of contact systems. Our EDS is already known in dimensions 2 and 3, where it was used by Griffiths in applications to mechanical problems and Lagrangian systems. It is also known in any dimension but just for flat Euclidean space. Having found the Lagrangian forms \(\alpha _{i} \in \varOmega ^{n}\), 0 ≤ i ≤ n, we are led to the associated functionals \(\mathcal{F}_{i}(N) =\int _{N}\alpha _{i}\), on the set of hypersurfaces N ⊂ M, and to their Poincaré-Cartan forms. A particular functional relates to scalar curvature and thus we are confronted with an interesting new equation.

Keywords

Contact Structure Contact Manifold Constant Sectional Curvature Tangent Sphere Tangent Sphere Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. PIEF-GA-2012-332209.

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Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Dipartimento di Matematica “Giuseppe Peano”Università degli Studi di TorinoTorinoItaly
  2. 2.Centro de Investigação em Matemática e AplicaçõesUniversity of ÉvoraÉvoraPortugal

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