I have questions referred to the second fundamental form on riemannian manifolds and mean curvature:
Is there a notion of mean curvature for submanifolds with arbitrary codimension?
I couldn't find something in the net. But I have herad about the notion of second fundamental form (=II) for arbitrary submanifolds. Maybe one can define it as the trace of II? On the other hand, as I can see, the second fundamental form II depends on the choice of a normal vector $\nu$. Is this true?
For instance, I'm faced with the following situation:
We consider an isometric immersion $f:M\rightarrow \mathbb{R}^m$, s.t. M is a submanifold of dimension n. Then we take the function $\nu:M\rightarrow \mathbb{R}$, which assigns to every point $p\in M$ the trace of its second fundamental form. Is the choice of second fundamental form unique yet?
Regards