Let $X : U \rightarrow R^n$ a vector field such that
$ \langle X(p); p \rangle = 0 \,\,\,\,\,\,\,\,\ \forall p \in R^n; \,\,\,\,\,\ ||p|| = 1$;
where $\langle \cdot ; \cdot \rangle$ is the inner product canonic in $R^n$ and $||\cdot ||$ is the Euclidian norm. show that any orbit that intercepts the unitary sphere
$S^{n-1} := \{p \in R^n; ||p|| = 1 \}$
is contained in the sphere.
I've tried several local theorems to use, but none seems to work well. Can anyone help?
Lets consider the integral curve $\gamma: I \to \mathbb R^n$ of the vector field $X$, where $I$ is an interval. That is, $\gamma'(t) = X(\gamma(t))$ for all $t\in I$. Suppose $|\gamma(t_0)|=1$ for some $t_0 \in I$. We want to show that $|\gamma(t)|=1$ for all $t\in I$.
Consider $f(t) = \langle \gamma(t),\gamma(t)\rangle$.
We have to show that $f(t) = 1$ for all $t\in I$. Notice that $f'(t) = 2\langle \gamma'(t), \gamma(t) \rangle$. Since $\gamma'(t) = X(\gamma(t))$ we have $$f'(t) = \langle X(\gamma(t)),\gamma(t) \rangle,$$ which is $0$ by hypothesis. So we have $f' = 0$ on the interval $I$ and therefore $f$ is constant. Since $f(t_0)=1$, we conclude $f(t)= 1$ for all $t\in I$. Therefore, $\gamma(t)\in \mathbb S^{n-1}$ for all $t\in I$.