- #1
Silviu
- 624
- 11
Hello! I am not sure I understand the idea of vector field on a manifold. The book I read is Geometry, Topology and Physics by Mikio Nakahara. The way this is defined there is: "If a vector is assigned smoothly to each point on M, it is called a vector field over M". Thinking about the 2D Euclidean space, it would look like at each point there is a vector (tangent to a curve inside that manifold) pointing in a certain direction, going smoothly from a point to another. However, he says then that "each component of a vector field is a smooth function from M to R". I am not sure I understand why is this. The components of a vector field are the vectors and they map from F(M) to R, where F(M) is the set of all smooth functions on M and F(M) is not the same thing as M. Also he says that "A vector field X at p ##\in## M is denoted by ##X|_p##, which is an element of ##T_pM##". I am confused here, too. By the first definition, a vector field associates to each point in M a vector, in a smooth way. How can you have a vector field at a point, when you have just one vector at that point (I understand you can define the tangent vector space for every point, but this is not what we describe here, right?). Can someone clarify these for me please? Thank you!