Global differential geometry by S. S. Chern

By S. S. Chern

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In coordinates, is described by giving r functions of n variables Xl, . . , xn y l , . . , yr x x). ,r or simply y F(x). We will frequently use the more dangerous notation y y(x). Let Yo F(xo); the Jacobian matrix (ay'" jax i )(xo) has the following significance. = = 1, = yr v = x(O) x" W = Y (O) = F. v y(t) = F(x(t)) F - image of jR" under F �_____________ y r - I 4------ x l , . . 3 Xo. x(t) such that x (O) Xo x(t) Xo + tv . V (0) = L � (xo) x i (0) ax i=l 11 = L i= l (� a "' ) (xo)v i ax The assignment v � w is, from this expression, independent of the curve and defines a linear transformation, the differential of at F Xo F ..

We shall soon define a "manifold" to be a space that, like the surface of the Earth, can be covered by a family of local coordinate systems. A manifold will turn out to be the most general space in which one can use differential and integral calculus with roughly the same facility as in euclidean space. It should be recalled, though, that calculus in ]R 3 demands special care when curvilinear coordinates are required. The most familiar manifold is N-dimensional euclidean space ]R N , that is, the space of ordered N tuples (x I .

In fact, it is shown in every topology book that any s u bs et X of �n (with the induced topology) is compact if and only if X 1. 2. X is a closed subset of lRn , is a bounded subset, that is, II x II < some number c, for all x E X. Finally we shall need two properties of continuous maps. First The continuous image of a compact space is itself compact. If f : G ---+ M is continuous and if { V; } is an open cover of f (G ) c M, then { f I (Vi ) } is an open cover of G. Since G is compact we can extract a finite open subcover { f- I (Va ) } of G , and then { Va } is a finite subcover of f(G).

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