ONT Re: Differential And Riemannian Manifolds
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DARM. Note 10
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And now the real fun begins ...
Tangent Vectors and Tangent Spaces
| Let X be a manifold of class C^p (p >= 1). Let x be a point of X.
| We consider triples (U, q, v) where (U, q) is a chart at x and v is
| an element of the vector space in which qU lies. We say that two such
| triples (U, q, v) and (V, r, w) are "equivalent" if the derivative of
| rq^-1 at qx maps v on w. The formula reads:
|
| (rq^-1)'(qx)v = w
|
| (obviously an equivalence relation by the chain rule).
|
| An equivalence class of such triples is called a "tangent vector" of X at x.
| The set of such tangent vectors is called the "tangent space" of X at x and
| is denoted by "T_x (X)". Each chart (U, q) determines a bijection of T_x (X)
| on a Banach space, namely the equivalence class of (U, q, v) corresponds to
| the vector v. By means of such a bijection it is possible to transport to
| T_x (X) the structure of topological vector space given by the chart, and
| it is immediate that this structure is independent of the chart selected.
|
| Lang, DARM, pp. 25-26.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
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