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Rotation Minimizing Vector Fields and Frames in Riemannian Manifolds

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Book cover Geometry, Algebra and Applications: From Mechanics to Cryptography

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 161))

Abstract

We prove that a normal vector field along a curve in \(\mathbb {R}^{3}\) is rotation minimizing (RM) if and only if it is parallel respect to the normal connection. This allows us to generalize all the results of RM vectors and frames to curves immersed in Riemannian manifolds.

Dedicated to Jaime Muñoz Masqué, with a deep gratitude for his generous assistance in the earlier years of my career, on the occasion of his 65th birthday

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Acknowledgments

The author wants to express his gratitude to his colleagues Marco Castrillón, Laureano González-Vega, Bert Jütler and Gema Quintana for the useful talks about the theory of RM vectors and frames.

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computation, University of Cantabria, Avda. de Los Castros, s/n, 39071, Santander, Spain

    Fernando Etayo

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  1. Fernando Etayo
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Correspondence to Fernando Etayo .

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Editors and Affiliations

  1. Departamento de Geometría y Topología, Universidad Complutense de Madrid, Madrid, Spain

    Marco Castrillón López

  2. Physical and Information Technology, Spanish National Research Council (CSIC), Madrid, Spain

    Luis Hernández Encinas

  3. Institute of Fundamental Physics, Spanish National Research Council (CSIC), Madrid, Spain

    Pedro Martínez Gadea

  4. Departamento de Matemática Aplicada, Universidad Politécnica de Madrid, Madrid, Spain

    Mª Eugenia Rosado María

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Etayo, F. (2016). Rotation Minimizing Vector Fields and Frames in Riemannian Manifolds. In: Castrillón López, M., Hernández Encinas, L., Martínez Gadea, P., Rosado María, M. (eds) Geometry, Algebra and Applications: From Mechanics to Cryptography. Springer Proceedings in Mathematics & Statistics, vol 161. Springer, Cham. https://doi.org/10.1007/978-3-319-32085-4_8

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