In symplectic geometry, the Grassmannian of isotropic planes for a symplectic vector space is a well known and well studied object; for example, one can realize it as a homogeneous space with a known stabilizer subgroup of the symplectic group, and one can also realize a Schubert cell decomposition.


However, one can relax the symplectic condition in two ways: by relaxing its degeneracy condition, and by allowing it to take vector values (i.e., allowing for general tensors). A good example generalizing both of these situations can be found from the study of distributions: for any given distribution $D$ on a tangent bundle $TM$ of some smooth manifold $M$, there is a natural (O'Neill?) torsion tensor $T \in \Gamma(\wedge^2D^\* \otimes TM/D)%$ defined by: 


\begin{equation}
T(X,Y) = [X,Y] \text{ mod }D
\end{equation}


Clearly, $T$ is a skew symmetric bilinear map. However, its degeneracy could vary depending on the distribution. On one hand, $T = 0$ identically gives that $D$ is integrable. On the other, it being nondegenerate gives that $D$ is maximally non-integrable and is a contact structure. One can still make sense of planes isotropic with respect to $T$, as the planes on which $T$ vanish identically.



> 
> Is there anything known about the isotropic Grassmannian of planes
>  with respect to the above tensor (e.g., is it homogeneous with respect
>  to a nice known Lie group? Can one describe its Schubert cell
>  decomposition?)? With respect to more general settings/tensors as above?
> 
> 
> 


I imagine in the case of the above particular tensor, one treats each component of the vector as a (possibly degenerate) symplectic form but I have not seen anything about it in the literature, so I'm hesitant to declare everything should be *exactly* the same as for the standard Grassmannian of isotropic planes with respect to an ordinary symplectic form.



Grassmannians of planes isotropic with respect to general tensors
What you are asking about is very classical in the theory of exterior differential systems. The subspaces of $D$ that you are calling `isotropic' are what Élie Cartan called the *integral elements* of the differential ideal $\mathcal{I}$ generated by the sections of $D^\perp\subset T^\*M$ (the annihilator subbundle of $D$). The geometry and topology of these spaces of integral elements can be quite subtle, and a significant part of the basic theory of exterior differential systems is devoted to developing ways to describe these subspaces of the Grassmann bundles, particularly deriving necessary and sufficient conditions that a given integral element $E\subset D\_x\subset T\_xM$ should be a tangent space of an *integral manifold*, i.e., a submanifold $N\subset M$ all of whose tangent spaces are sub-planes of the plane field $D$. This leads to the Cartan-Kähler theory and beyond.


For references, you can look in our book *Exterior Differential Systems* (Bryant, et. al) or in *Cartan for Beginners* (Ivey and Landsberg). 


