Return to Question

Edit: According to the comment of Prof. Bryant, I revise my question.

Is there a reasonable and non trivial (geometric) interpretation for the following quantity on a compact Riemannian manifold $(M,g)$ of dimension $3$

$$q=\sup_{\alpha}\int_M \alpha \wedge d\alpha $$

where $\sup$ is taken over all $1$-forms $\alpha_X$ associated to the unit length vector field $X$ on $M$. that is $\alpha_X (Y)=g(X,Y)$

Edit: According to the comment of Prof. Bryant, I revise my question.

Is there a reasonable and non trivial (geometric) interpretation for the following quantity on a compact Riemannian manifold $(M,g)$ of dimension $3$

$$q=\sup_{\alpha}\int_M \alpha \wedge d\alpha $$

where $\sup$ is taken over all $1$-forms $\alpha$ of $g$-length $1$.

Is there a reasonable and non trivial (geometric) interpretation for the following quantity on a compact Riemannian manifold $(M,g)$ of dimension $3$ whose Euler characteristic is zero:

$$q=\sup\int_M \alpha \wedge d\alpha $$

where $\sup$ is taken over all $1$-forms associated to the unit length vector field $X$ on $M$. that is $\alpha (Y)=X._g Y$

One can consider the same quantity for a compact Riemanian manifold of dimension $3$, with arbitrary Euler characteristic, but take $\sup$ over all vector fields whose length is at most $1$.

Edit: According to the comment of Prof. Bryant, I revise my question.

Is there a reasonable and non trivial (geometric) interpretation for the following quantity on a compact Riemannian manifold $(M,g)$ of dimension $3$

$$q=\sup_{\alpha}\int_M \alpha \wedge d\alpha $$

where $\sup$ is taken over all $1$-forms $\alpha_X$ associated to the unit length vector field $X$ on $M$. that is $\alpha_X (Y)=g(X,Y)$