I am facing the following problem:
Given two one-dimensional submanifolds $M_1,M_2$ (with non-empty intersection) of some manifold $M$, can one define an angle between $M_1$ and $M_2$?
I know how to compute angles of vectors in tangent spaces (via inner product), but I do not know how to extrapolate this to the manifold. In none of the references I've checked they do such construction.
This question is kind of related to this one, in the sense that I am trying to compute the angle between $N_1$ and $N_2$, if this even makes sense.
I'm a second year student and I'm also studying differential geometry. I tried to write my idea/intuition about the question done. If it's wrong I'll delete it.
Suppose you have two sets $M_1$ and $M_2$ in $\mathbb R^n$ and two smooth maps $f_1:\mathbb R^n\to \mathbb R^k$ and $f_2:\mathbb R^n\to \mathbb R^k$.
If $f_1^{-1}(\textbf a_1)=M_1$ and $f_2^{-1}(\textbf a_2)=M_2$, with $f_1$ and $f_2$ submersions (which means that $(f_1)_*$ and $(f_2)_*$ are surjective), than we have that $M_i\subset\mathbb R^n$ are embedded submanifolds, with dimension equal to $\text {codim}(\mathbb R^k)$.
(if $\exists \bar x\in\mathbb R^n:rk(f_i)(\bar x)<\dim(M_i)$, you should check that $\bar x\notin f_i^{-1}(\textbf a_i)$)
Using the fact that $rk(f_i\vert_{M_i})=\dim(M_i)$ you could compute, for $p\in M_1\cap M_2$, (in general in the intersection of two local charts of the two submanifolds) the tangent space $T_p(M_i)\cong\ker(f_i)_{*p}$ and find the angle between tangent vectors of the two tangent spaces.