Differentiable Manifold

Daizhan Cheng (), Xiaoming Hu () and Tielong Shen ()
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Daizhan Cheng: Chinese Academy of Sciences, Academy of Mathematics & Systems Science
Xiaoming Hu: Royal Institute of Technology, Optimization and Systems Theory
Tielong Shen: Sophia University, Department of Engineering and Applied Sciences

Chapter Chapter 3 in Analysis and Design of Nonlinear Control Systems, 2010, pp 47-89 from Springer

Abstract: Abstract This chapter provides an outline of Differential Geometry. First we describe the fundamental structure of a differentiable manifold and some related basic concepts, including mappings between manifolds, smooth functions, sub-manifolds. The concept of fiber bundle is also introduced. Then vector fields, their integral curves, Lie derivatives, distributions are discussed intensively. The dual concepts, namely, covector fields, their Lie derivatives with respect to a vector field, co-distributions and the relations with the prime ones are also discussed. Finally, some important theorems and formulas, such as Frobenius’ theorem, Lie series expansions and Chow’s theorem etc. are presented. This chapter provides a fundamental tool for the analysis of nonlinear control systems.

Keywords: Poisson Bracket; Coordinate Frame; Symplectic Manifold; Integral Curve; Differentiable Manifold (search for similar items in EconPapers)
Date: 2010
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-11550-9_3

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DOI: 10.1007/978-3-642-11550-9_3

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