From Frenet to Cartan : the method of moving frames
著者
書誌事項
From Frenet to Cartan : the method of moving frames
（Graduate studies in mathematics, v. 178）
American Mathematical Society, c2017
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注記
Includes bibliographical references (p. 397-402) and index
内容説明・目次
内容説明
The method of moving frames originated in the early nineteenth century with the notion of the Frenet frame along a curve in Euclidean space. Later, Darboux expanded this idea to the study of surfaces. The method was brought to its full power in the early twentieth century by Elie Cartan, and its development continues today with the work of Fels, Olver, and others.
This book is an introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in Euclidean space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces, including Euclidean, Minkowski, equi-affine, and projective spaces. Later chapters include applications to several classical problems in differential geometry, as well as an introduction to the nonhomogeneous case via moving frames on Riemannian manifolds.
The book is written in a reader-friendly style, building on already familiar concepts from curves and surfaces in Euclidean space. A special feature of this book is the inclusion of detailed guidance regarding the use of the computer algebra system Maple (TM) to perform many of the computations involved in the exercises.
目次
Background material: Assorted notions from differential geometry
Differential forms
Curves and surfaces in homogeneous spaces via the method of moving frames: Homogeneous spaces
Curves and surfaces in Euclidean space
Curves and surfaces in Minkowski space
Curves and surfaces in equi-affine space
Curves and surfaces in projective space
Applications of moving frames: Minimal surfaces in $\mathbb{E}^3$ and $\mathbb{A}^3$
Pseudospherical surfaces in Backlund's theorem
Two classical theorems
Beyond the flat case: Moving frames on Riemannian manifolds: Curves and surfaces in elliptic and hyperbolic spaces
The nonhomogeneous case: Moving frames on Riemannian manifolds
Bibliography
Index.
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