Titlebook: Differential Geometry and Mathematical Physics; Part I. Manifolds, L Gerd Rudolph,Matthias Schmidt Book 2013 Springer Science+Business Medi

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骨

Vector Bundles,n physical models may be viewed in a coordinate-free manner as sections of certain vector bundles. We start by observing that the tangent spaces of a manifold combine in a natural way into what is called the tangent bundle. By taking the properties of the tangent bundle as axioms, we arrive at the n

緊張過(guò)度

Vector Fields,gebra of smooth functions, discuss the notions of integral curve and flow, introduce the Lie derivative and give a brief account to time-dependent vector fields. Thereafter, we give an introduction to (geometric) distributions, i.e., subsets of the tangent bundle which are locally spanned by vector

Felicitous

Differential Forms,he theory of integration and the Stokes Theorem, as well as an introduction to de Rham cohomology. Next, we discuss elements of Riemannian geometry and Hodge duality. As an application, we show how classical Maxwell electrodynamics can be understood in a coordinate-free way using the language of dif

EWE

Lie Groups,lassical groups. Thereafter, we discuss left-invariant vector fields, define the Lie algebra of a Lie group and construct the exponential mapping, which provides a local diffeomorphism between the group and its algebra. This proves useful both for the study of the local structure of Lie groups and f

Hemiparesis

Lie Group Actions,fields of a special type, called Killing vector fields. We prove the Orbit Theorem, which states that the distribution spanned by the Killing vector fields is integrable and that its integral manifolds coincide with the connected components of the orbits of the action. This way, every orbit gets end

Hemiparesis

Linear Symplectic Algebra,mplectic geometry provides the natural mathematical framework for the study of Hamiltonian systems. In this chapter, we present linear symplectic algebra. We start with a discussion of the elementary properties of symplectic vector spaces, the various types of their subspaces and linear symplectic r

amorphous

Symplectic Geometry, are locally equivalent. Thus, in sharp contrast to the situation in Riemannian geometry, symplectic manifolds of the same dimension can at most differ globally. The second important observation is that a symplectic structure provides a duality between smooth functions and certain vector fields, cal

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