There is something I am confused about. I am reading Loring An introduction to differential geometry. I am confused why if $F(p) = F(q) = z$, they then have the same push-forward of the vector field. Is it because we are dealing with the same target space. From first inspection:
For $F : N \rightarrow M \in C^{\infty}(M, N)$ map of manifolds we have:
$$F_{*}(X_p)f = X_p(f \circ F)$$
$$F_{*}(X_q)f = X_q(f \circ F)$$
Why are they the same ? I see the picture that Loring posted in his book and it makes sense to me. Though I don't see the rigor involved.
Best Answer
They are not necessarily the same and that's exactly the problem. If you have a smooth map $F \in C^\infty(M, N)$ that is not injective, say $F(p) = F(q) = z$, there might not be a vector fields $Y$ on $N$ that is $F$-related to a vector fields $X$ on $M$, i.e. $$F_{*s}(X_s) = Y_{F(s)}, \quad \forall s \in M.$$ Indeed $Y_{F(p)} = Y_{F(q)} = Y_z$ is not well-defined since $F_{*p}(X_p)$ and $F_{*q}(X_q)$ may not be the same tangent vector.