1.$M$ is a connected manifold, dim$M \geq 2$, $f:M \rightarrow \mathbb{R}$ is smooth, then $f$ is not an injection.
2.$M, N$ are two manifolds, and $M$ is connected, $f:M \rightarrow N $ is smooth,if for any $p\in M$, $f_*p=0$, then $f$ is constant.
For the first question, $f(M)$ is connected, so $f(M)=R$. If $f$ is an injection, then $f$ is a smooth bijection. And then?
For the second question, I don't know how to use the condition the connectedness.
Thanks a lot.
$Im(f)$ is connected, which means it's an interval of $\mathbb{R}$, possibly not the whole of $\mathbb{R}$ as you have written. Now, note that $f_{*}$ cannot be trivial, for then by Problem 2, $f$ loses injectivity (connectedness of $M$ is used). Thus $\exists$ $p\in{M}$ such that $f_{*}:T_{p}M\to\mathbb{R}$ is not the $0$ map. But then $f_{*}$ has to be surjective at that point, implying that $f$ is a submersion at that point. But an injective submersion is impossible if $dim(M)\geq 2$ !
Transfer the question locally onto open subsets of euclidean spaces by using the "manifold" structure of $M$ and $N$. Then you know that any smooth function that has trivial push-forward has to be constant (in euclidean space). Thus, you arrive at the conclusion that the given function $f$ must be locally constant. Now can you use connectedness ?
