Notes on Differential Geometry and Lie Groups

However, for any point p on the manifold M and for any chart whose domain contains p, there is a convenient basis of the tangent space Tp(M). The third definition is also the most convenient one to define vector fields. A few technical complications arise when M is not a smooth manifold (when k 6=∞), but these are easily overcome using “stationary germs.” As pointed out by Serre in [161] (Chapter III, Section 8), the relationship between the first definition and the third definition of the tangent space at p is best described by a nondegenerate pairing which shows that Tp(M) is the dual of the space of point derivations at p that vanish on stationary germs. This pairing is presented in Section 7.4. The most intuitive method to define tangent vectors is to use curves. Let p ∈M be any point on M and let γ : (− , )→M be a C-curve passing through p, that is, with γ(0) = p. Unfortunately, if M is not embedded in any R , the derivative γ′(0) does not make sense. However, for any chart, (U,φ), at p, the map φ◦γ is a C-curve in R and the tangent vector v = (φ ◦ γ)′(0) is well defined. The trouble is that different curves may yield the same v! To remedy this problem, we define an equivalence relation on curves through p as follows: Definition 7.6. Given a C manifold, M , of dimension n, for any p ∈ M , two C-curves, γ1 : (− 1, 1)→M and γ2 : (− 2, 2)→M , through p (i.e., γ1(0) = γ2(0) = p) are equivalent iff there is some chart, (U,φ), at p so that (φ ◦ γ1)(0) = (φ ◦ γ2)(0).