TITLE: Curvature of nonsymmetric metric tensors?
QUESTION [6 upvotes]: Consider a smooth manifold $M$ of arbitrary dimension. We have notions of psuedo-Riemannian or Riemannian metrics on a manifold, and they differ in the slightest way of being positive-definite or not. However, what happens if we drop positive-definite AND symmetry? For example, if we had a nondegenerate bilinear form $g_p: T_p M \times T_p M \to \mathbb{R}$ that varied smoothly between points. Has this been explored in depth? It appears to me at surface level that one could still concoct connections, curvature, and possibly a notion of parallel transport in this flavor of 'smooth geometry'. A motivation for me to ask is as follows. Suppose $R$ is an $S$-algebra where $\Omega_{R/S}$ is reflexive and the canonical isomorphism $\phi: \Omega_{R/S} \to \Theta_{R/S}$ is an isomorphism of $R$-modules (i.e. nonsingular varieties). There exists a canonical map $\Omega_{R/S} \times \Theta_{R/S} \to R$, which is $R$-bilinear and nondegenerate, and is given by $\langle \omega, V \rangle = l(\omega)$ where $l:\Omega_{R/S} \to k$ such that $l \circ d_{R/S} = V$. This induces a morphism $$\Theta_{R/S} \times \Theta_{R/S} \xrightarrow{\phi^{-1} \times 1}\Omega_{R/S} \times \Theta_{R/S} \to R.$$ Natural questions that arise are is this composition $R$-bilinear nondegenerate, and when is it symmetric? Which symmetric $R$-bilinear forms factor through $\phi^{-1} \times 1$? When we work with a manifold and have a metric tensor that is bilinear and nondegenerate, just how interesting is this flavor of curvature (whatever it is supposed to mean)?

REPLY [7 votes]: A few remarks:  
First, in a sense, (special cases of) this (are) is very commonly studied.  Because a bilinear differential form $g$ as you have defined it can naturally be written as a sum $g = \sigma + \alpha$ where $\sigma$ is symmetric and $\alpha$ is skew-symmetric, you are, equivalently, asking about the geometry of the pair $(\sigma,\alpha)$.  The most famous example is that of Kähler geometry, i.e., where $\sigma$ is positive definite and $\alpha$ is nondegenerate and parallel with respect to the Levi-Civita connection of $\sigma$, but there are many variations on this (pseudo-Kähler, Hermitian, unitary...) that essentially have the same nature.  In general, when $\sigma$ is non-degenerate, you know that there is the canonical Levi-Civita connection of $\sigma$, but, of course, depending on the assumptions you make about $\alpha$, there could be other `canonical' connections. 
Second, it is not clear how you ought to define 'non-degenerate' for a general $g$, because there are different notions, and it might well be that what you want to call 'non-degenerate' depends on the applications you have in mind.  For example, if $M=\mathbb{R}^2$, and $g = \mathrm{d}x\otimes\mathrm{d}y$, then $g$ is 'degenerate' in the naïve sense (since it has tensor rank $1$ instead of $2$), but both $\sigma = \tfrac12(\mathrm{d}x\otimes\mathrm{d}y+\mathrm{d}y\otimes\mathrm{d}x)$ and $\alpha=\tfrac12(\mathrm{d}x\otimes\mathrm{d}y-\mathrm{d}y\otimes\mathrm{d}x)$ are non-degenerate in the usual senses for symmetric and anti-symmetric quadratic differential forms.  In particular, since $\sigma$ has a 'functorial' connection (being non-degenerate), $g$ does also.  You'd want to count such a $g$ as 'non-degenerate', no?
Finally, as Dmitry has already remarked, you do need some hypotheses beyond simple algebraic 'non-degeneracy' in order to have a 'functorial' (i.e., invariant under all diffeomorphisms)construction of a connection.  Such considerations will enter into any discussion of the right hypotheses, as Vit's discussion above makes clear.  I'll just add that sometimes it's important to take higher order derivatives of $g$ into consideration in order to define a class of structures for which a 'functorial' connection exists.  For example, as Dmitry pointed out, a closed non-degenerate $2$-form on $M$ does not determine a 'functorial' affine connection on $M$.  However, if you drop the assumption 'closed' and replace it with an appropriate higher order non-degeneracy condition, there sometimes is an associated functorial connection.  For example, there is an open set $\mathcal{F}$ of germs of $2$-forms on $4$-manifolds that is perserved by (local) diffeomorphisms that does have the property that any $2$-form $\alpha$ on $M^4$ whose germ at every point belongs to $\mathcal{F}$ possesses a functorial torsion-free connection $\nabla^\alpha$, but the definition of $\mathcal{F}$ (and of $\nabla^\alpha$) depends on higher derivatives of $\alpha$ than just first derivatives.  (In particular, $\mathrm{d}\alpha$ will be nowhere vanishing if $\alpha$ has its germs belonging to $\mathcal{F}$ at every point.)
Remark:  I was asked about how the above $\mathcal{F}$ is defined and how it works.  Here's a quick sketch of the result of applying Cartan's method of equivalence to this question:  
Start with a non-degenerate $2$-form $\alpha$ on a $4$-manifold $M$.  First, there exists a unique $1$-form $\beta$ on $M$ such that $\mathrm{d}\alpha = \beta\wedge\alpha$.  Let $\gamma = \mathrm{d}\beta$.  Then $0 = \mathrm{d}(\beta\wedge\alpha) = \gamma\wedge\alpha$.  Second, there exists a uniqe function $F$ on $M$ such that $\gamma^2 = F\,\alpha^2$, and the first condition defining $\mathcal{F}$ is that $F$ should be nowhere vanishing. For simplicity, I'm going to continue the analysis under the assumption that $F<0$ (there is a similar branch when $F>0$, but I'll leave that to the interested reader).  Set $F = -f^2$ where $f>0$.  Then, using $\gamma\wedge\alpha= 0$ and $\gamma^2 = -f^2\,\alpha^2$, we see that $(\gamma\pm f\alpha)^2 = 0$, but $(\gamma+f\alpha)\wedge(\gamma-f\alpha) = -2f^2\,\alpha^2\not=0$, so, setting $\gamma\pm f\,\alpha = \pm 2f\,\alpha_{\pm}$, we have
$$
\alpha = \alpha_+ + \alpha_-\quad\text{and}\quad
\gamma = f\,(\alpha_+ - \alpha_-),
$$
where $\alpha_\pm$ are a pair of (nonvanishing) decomposable $2$-forms whose wedge product is nonvanishing.
Finally, there are unique decompositions
$$
\beta = \beta_+ + \beta_- \quad\text{and}\quad \mathrm{d}f = \phi_+ + \phi_-
$$
where $\beta_\pm \wedge\alpha_{\pm} = \phi_\pm \wedge\alpha_{\pm} = 0$, and so there will be unique functions $g_\pm$ such that
$$
\beta_\pm\wedge\phi_\pm = g_\pm\,\alpha_\pm\,.
$$
The final 'open' conditions on $\alpha$ needed to define $\mathcal{F}$ are that $g_+$ and $g_-$ be nonvanishing.  
In this case, the $1$-forms $\beta_+$, $\beta_-$, $\phi_+$, and $\phi_-$ define a coframing on $M$ that is functorially associated to $\alpha$.  Once one has such a 'canonical' coframing, it is easy to define a connection, in fact, a large family of connections, on $M^4$ (for example, one of these connections (with torsion) will make the given coframing parallel) including some that are torsion-free.