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Kinematical Aspects of Levi-Civita Transport of Vectors and Tensors Along a Surface Curve

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The Mechanics of Ribbons and Möbius Bands

Abstract

The concept of parallelism along a surface curve, which was introduced by Levi-Civita in the context of n-dimensional Riemannian manifolds, is re-examined from a kinematical viewpoint. A special type of frame, whose angular velocity is determined by the rate at which the tangent plane turns as one moves along a surface curve, is defined and is called a Levi-Civita frame. The surface may be orientable or not. Vectors and tensors fixed on Levi-Civita frames are parallel transported. Covariant differentiation of vectors and tensors along a surface curve can be expressed in terms of the corresponding corotational rates measured on Levi-Civita frames. Relevant results on ruled surfaces are also included.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of California, Berkeley, CA, USA

    James Casey

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  1. James Casey
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Correspondence to James Casey .

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Editors and Affiliations

  1. Aeropsace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota, USA

    Roger Fosdick

  2. Okinawa Institute of Science and Technol Mathematical Soft Matter Unit, Okinawa, Japan

    Eliot Fried

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Casey, J. (2016). Kinematical Aspects of Levi-Civita Transport of Vectors and Tensors Along a Surface Curve. In: Fosdick, R., Fried, E. (eds) The Mechanics of Ribbons and Möbius Bands. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7300-3_12

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  • DOI: https://doi.org/10.1007/978-94-017-7300-3_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-017-7299-0

  • Online ISBN: 978-94-017-7300-3

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