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

  • James CaseyEmail author
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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.

Keywords

Levi-Civita parallelism Parallel transport Differential geometry of surfaces Adapted frames Covariant differentiation Ruled surfaces Developable surfaces Möbius band 

Mathematics Subject Classification

53A05 53A17 53A45 

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of CaliforniaBerkeleyUSA

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