Is there a name for this constant associated to smooth maps between spheres? (not degree)

Consider the following constant associated to smooth maps $F: S^{2n-1} \to S^n$ for $n \geq 2$:

Let $\omega \in \Omega^n(S^n)$ be a volume form with $\int_{S^n} \omega = 1$. Then there exists $\eta \in \Omega^{n-1}\left( S^{2n-1} \right)$ with $d\eta = F^*\omega$; define $$ h(F) = \int_{S^{2n-1}} \eta \wedge d\eta. $$

On a recent exam, I was given this definition and asked to show it is independent of choices of $\omega$ and $\eta$, invariant under smooth homotopy, and satisfies $h(F) = 0$ for odd $n$.

Is there a name for this constant? Does it have any important properties or show up in any interesting places?

Thanks!

Solution 1:

As Eric says, this integral equals the Hopf Invariant. This equality was first proven by Whitehead in

Whitehead, J. H. C., An expression of Hopf’s invariant as an integral, Proc. Natl. Acad. Sci. USA 33, 117-123 (1947). ZBL0030.07902.

when $n=2$ and then by Whitney in full generality; see sections 31, 33 in Chapter IV of

Whitney, Hassler, Geometric integration theory, Princeton Mathematical Series. Princeton, N. J.: Princeton University Press; London: Oxford University Press. XV, 387 p. (1957). ZBL0083.28204.