# Vector Fields and Vector Bundles

by AiRAVATA
Tags: bundles, fields, vector
 P: 173 I need help solving the following problem: Let $M,N$ be differentiable manifolds, and $f\in C^\infty(M,N)$. We say that the fields $X\in \mathfrak{X}(M)$ and $Y \in \mathfrak{X}(N)$ are f-related if and only if $f_{*p}(X(p))=Y_{f(p)}$ for all $p\in M$. Prove that: (a) $X$ and $Y$ are f-related if and only if $X(g \circ f)=Y(g) \circ f$, for all $g\in C^\infty(M)$. (b) If $X_i$ is f-related with $Y_i$, $i=1,2$, then $[X_1,X_2]$ is f-related with $[Y_1,Y_2]$. I know this is silly, but my main problem is that i dont know how the identity $f_{*p}(X(p))=Y_{f(p)}$ looks like. What i mean is the following: If $X=\sum_{i=1}^m X_i \frac{\partial }{\partial x_i}$, and $Y=\sum_{j=1}^m Y_j \frac{\partial }{\partial y_j}$, then $$f_{*p}(X(p))=f_{*p}(\sum_{i=1}^m X_i(p) \frac{\partial }{\partial x_i})=\sum_{i=1}^m f_{*p}(X_i(p) \frac{\partial }{\partial x_i})$$ so $$f_{*p}(X(p))=\sum_{i=1}^m f_{*p}(X_i(p))\frac{\partial }{\partial x_i}+X_i(p) f_{*p}(\frac{\partial }{\partial x_i})=\sum_{j=1}^m Y_j (f(p))\frac{\partial }{\partial y_j}.$$ Is this correct? Another question. Does anyone know some good online notes regarding vector bundles?