For much ofphysics and mathematics the concept ofa continuous map, provided by topology, is not sufficient. What is often required is a notion ofdifferentiable or smooth maps between spaces. For this, our spaces will need a structure something like that ofa surface in Euclidean space . The key ingredient is the concept ofa differentiable manifold, which can be though of as topological space that is ‘locally Euclidean’ at every point. Differential geometry is the area of mathematics dealing with these structures. Of the many excellent books on the subject, the reader is referred in particular to [1–14].
Think ofthe surface ofthe Earth. Since it is a sphere, it is neither metrically nor topologically identical with the Euclidean plane . A typical atlas ofthe world consists ofseparate pages called charts, each representing different regions of the Earth. This representation is not metrically correct since the curved surface of the Earth must be flattened out to conform with a sheet of paper, but it is at least smoothly continuous. Each chart has regions where it connects with other charts part of France may find itself on a map of Germany, for example – and the correspondence between the charts in the overlapping regions should be continuous and smooth. Some charts may even find themselves entirely inside others; for example, a map of Italy will reappear on a separate page devoted entirely to Europe. Ideally, the entire surface of the Earth should be covered by the different charts of the atlas, although this may not strictly be the case for a real atlas, since the north and south poles are not always properly covered by some chart. We have here the archetype of a differentiable manifold.
Points of will usually be denoted from now on by superscripted coordinates, . In Chapter 12 we defined a function to be if all its partia derivatives
exist and are continuous for function is simply a continuous function, while a function is one that is for all values of such a function wil generally be referred to simply as a differentiable function. A differentiable function need not be analytic (expandable as a power series in a neighbourhood ofany point), as illustrated by the function
which is differentiable but not analytic at since its power series would have al coefficients zero at
A map is said to be when expressed in coordinates
each of the real-valued functions . Similarly, the notion of differentiable and analytic functions can be extended to maps between Euclidean spaces of arbitrary dimensions.
Contents
- 15.1 Differentiable manifolds
- 15.2 Differentiable maps and curves
- 15.3 Tangent, cotangent and tensor spaces
- 15.4 Tangent map and submanifolds
- 15.5 Commutators, flows and Lie derivatives
- 15.6 Distributions and Frobenius theorem
Concept index
Terms by section. Links lead to the brown underlined definitions in the Chinese reading text.
15.1 Differentiable manifolds
- topological manifold拓扑流形
- coordinate chart坐标卡
- coordinate neighbourhood坐标邻域
- coordinate map坐标映射
- coordinate function坐标函数
- transition function过渡函数
- atlas图册
- maximal atlas极大图册
- differentiable structure可微结构
- differentiable manifold可微流形
15.2 Differentiable maps and curves
15.3 Tangent, cotangent and tensor spaces
- tangent vector切向量
- tangent space切空间
- cotangent space余切空间
- vector field向量场
- covector field余向量场
- tensor field张量场
- tangent bundle切丛
- cotangent bundle余切丛
15.4 Tangent map and submanifolds
15.5 Commutators, flows and Lie derivatives
- commutator对易子
- Lie bracket李括号
- integral curve积分曲线
- orbit轨道
- complete vector field完备向量场
- local flow局部流
- Lie derivative李导数