Is there a notion of mean curvature for general submanifolds? 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
 A: Yes, the second fundamental form does depend on the choice of outward normal. For a general codimension $1$ submanifold, choose a normal vector (i.e. $\nabla \tau$ where $\tau$ is the distance function to the boundary) and then define the second fundamental form
$$S(X, Y) = g(\nabla_X \nabla \tau, Y)$$
The mean curvature is then the trace of this form. If the submanifold has codimension $k$, then there will be $k$ linearly independent normal vectors to your submanifold, for each of these you may define a second fundamental form as above and then take the trace, to obtain a version of mean curvature in the direction of each particular normal.
If you like, you can then combine all of these mean curvatures into a "mean curvature vector." More specifically, if $\nu_1, \ldots, \nu_k$ are your choices of unit normals to your submanifold, and $H_i$ is the mean curvature w.r.t $\nu_i$ as defined above, you may consider the vector $$\hat H = \sum H_i \nu_i$$
Then it follows that $g(\hat H, \nu_k) = H_k$ so that in this sense the mean curvature normal has the property that $g(\hat H, X)$ is "the mean curvature in the direction $X$".
Regarding the second part of your question, you need to properly interpret the phrase "trace of the second fundamental form" in the context I have described above.
