Homotopy class of maps into Stiefel manifolds Motivation
Hopf theorem, asserts that $C^0$-maps $f:M^n\to \mathbb{S}^n$  from an orientable, closed n-manifold into an n-sphere are classified up to homotopy by their degree $deg(f)$.
The theorem not only says that $[\mathbb{S}^n, \mathbb{S}^n] \simeq \mathbb{Z} $ but also gives us a way to compute the complexity of the map, namely the degree.
I am looking for a similar invariant of maps into Stiefel manifolds and orthogonal groups (they should be related).

1)Consider a map $f:\mathbb{S}^n \to V_k(\mathbb{R}^N)$ where $V_k(\mathbb{R})^n$ is the Stiefel manifold of $k$-orthogonal frames of $\mathbb{R}^N$. 
  Is there an invariant $\mathcal{I}(f)$ that similarly to the degree, provides us with a correspondence with the homotopy classes of maps $[\mathbb{S}^n , V_k(\mathbb{R}^N)]$?
2) What can we say with the orthogonal group $O(k)$ in place of $V_k(\mathbb{R}^N)$? (This should be related)

What we are looking for
Of course, if $N>k+1$ then $V_k(\mathbb{R}^N)$ is simple and  $[\mathbb{S}^n , V_k(\mathbb{R}^N)]\sim \pi_n  V_k(\mathbb{R}^N)\simeq \mathbb{Z} \text{ or } \mathbb{Z}/2$
but this is not enough, we need to pick a generator and once we have done this how do we associate to a function  a multiple of the generator?.
As the degree of $f:M^n\to \mathbb{S}^n$ can be defined homologically 
($f_*[M^n]= \deg(f)[\mathbb{S}^n]$), I expect that for our map $f:\mathbb{S}^n\to 
 V_k(\mathbb{R}^N)$  we can use  something like a set of integers
$\langle f_*[\mathbb{S}^n],[g_i]  \rangle\in \mathbb{Z}$ where $[g_i]\in H_*(V_k(\mathbb{R}^N))$.
 A: Maybe what you looking for is known under the name generalized curvatura integra (for the case $N> k+1$). I will formulate it not for $S^n$ but more generally for a $m$-dimensional framed manifold $M$, i.e. there is an embedding $F \colon M\to \mathbb R^{m+k}$ with trivialized normal bundle $\nu(F)\cong\varepsilon ^k$. This gives a map
$$
c\colon M \to V_k(\mathbb R^{m+k}),\quad p\mapsto \nu(F)_p\cong \mathbb R^k \subset T(\mathbb R^{m+k})_{F(p)}\cong \mathbb R^{m+k}.
$$
Definition: The generalized curvatura integra (gci) s defined by
$$
c_\ast[M] \in H_m(V_{m+k,k}) = 
\begin{cases}
\mathbb Z &m \equiv 0 \mod 2\\
\mathbb Z_2 & m\equiv 1 \mod 2.
\end{cases}
$$
Kervaire computed the gci in Relative characteristic classes and Courbure integrale generalisee et homotopie as follows:
$$
c_\ast[M] = H(M,F) +
\begin{cases}
\chi(M)/2 \in \mathbb Z, \,m\equiv 0\mod 2\\
\chi_{1/2}(M) \in \mathbb Z,\, m\equiv 1\mod 2\\
\end{cases}
$$
where 


*

*$\chi(M)$ is the Euler characteristic

*$\chi_{1/2}(M)$ is the Kervaire semicharacteristic and is defined as
$$
\chi_{1/2}(M)= \sum_{j=0}^{(m-1)/2} \dim_{\mathbb Z_2} H_j(M,\mathbb Z_2) \mod 2 \in \mathbb Z_2
$$

*$H(M,F)=0$ if $m \equiv 0 \mod 2$ and in case of $m \equiv 1\mod 2$ the number $H(M,F)$ is
defined as follows: Since $M$ is framed we have by the Pontryagin-Thom construction an induced
map $\tilde F \colon S^{k+m} \to S^k$. $H(M,F)$ is the determined by the Steenrod square in the mapping cone $X = S^k \cup_{\tilde F} D^{m+k+1}$, i.e. if $x \in H^k(X;\mathbb Z_2)$ and $y \in H^{k+m+1}(X;\mathbb Z_2)$ are generators then 
$$
Sq^{m+1}(x) = H(M,F)\cdot y.
$$
