I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain:
If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ then $T_xM\in G_n(\mathbb{R}^{d})$ for all $x\in M$. therefore I can check the closeness of two elements through $D$.
I wonder if I could do the same in the general case of a manifold $M$ of dimension $n$?
That is, if $M$ is manifold of dimension $n$ for Whitney embedding theorem exists $f:M\rightarrow f(M)\subset \mathbb{R}^{k}$ diffeomorphism. Then $d_xf:T_xM\rightarrow T_{f(x)}f(M)$ and I am tempted to define $$D(T_xM,T_yM):=D(d_xf(T_xM),d_yf(T_yM))$$
but this causes problems . I appreciate it if I could suggest a way to measure these tangent spaces.
Edit: I wonder if the fibers of a vector bundle $TM$ are continuous.
thanks for the help