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The Taylor expansion of the exponential map and geometric applications

Abstract

In this work we consider the Taylor expansion of the exponential map of a submanifold immersed in \(\mathbb {R}^n\) up to order three, in order to introduce the concepts of lateral and frontal deviation. We compute the directions of extreme lateral and frontal deviation for surfaces in \(\mathbb {R}^3.\) Also we compute, by using the Taylor expansion, the directions of high contact with hyperspheres of a surface immersed in \(\mathbb {R}^4\) and the asymptotic directions of a surface immersed in \(\mathbb {R}^n.\)

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Correspondence to M. G. Monera.

Additional information

This work was partially supported by DGCYT grant no. MTM2009-08933.

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Monera, M.G., Montesinos-Amilibia, A. & Sanabria-Codesal, E. The Taylor expansion of the exponential map and geometric applications. RACSAM 108, 881–906 (2014). https://doi.org/10.1007/s13398-013-0149-z

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  • DOI: https://doi.org/10.1007/s13398-013-0149-z

Keywords

  • Exponential map
  • Surfaces
  • Extremal directions
  • Contact
  • Normal torsion

Mathematics Subject Classification (2000)

  • 53A05
  • 53B20