This is a follow-up question to my previous one where I was trying to understand the classes of Legendrian immersions of circles into contact manifolds.

I'm interested in classifying isotropic immersions (of spheres, in my particular application, if that makes a difference) into contact manifolds up to regular homotopy through isotropic immersions. Let $(M,\xi)$ be a contact (2n+1)-manifold; here $\xi$ is a distribution of $2n$-planes in $TM$. An isotropic immersion of a manifold N (with dimension $k\leq n$, for $k=n$ this is a Legendrian immersion) is an immersion $f:N\rightarrow M$ where the image of $df_x:T_x N\rightarrow T_{f(x)}M$ lies in the plane $\xi_{f(x)}$ for all $x$.

Mike Usher's answer to my previous question directed me to a paper by Ekholm, Etnyre and Sullivan, in particular section 3.3 (p.19), which defines the rotation class of a Legendrian $f$ as follows.

If $\alpha$ is a contact 1-form on M, that is Ker $\alpha=\xi$, then $d\alpha_p|\xi_p$ is a symplectic form on $\xi_p$. If one chooses any complex structure $J$ for $\xi$ that is compatible with the symplectic structure, then complexifying $df$ to be $df_{\mathbb C}:TL\otimes \mathbb{C}\rightarrow\xi$ is a fiberwise bundle isomorphism.

The rotation class is the homotopy class of $(f,df_{\mathbb C})$ in the space of complex fiberwise isomorphisms $TL\otimes\mathbb{C}\rightarrow \xi$ and is denoted $r(f)$. An h-principle for Legendrian immersions implies that $r(f)$ is a complete invariant for $f$ up to regular homotopy through Legendrian immersions.

First, I can see how the above definition generalizes to isotropic immersions (namely, instead of bundle isomorphisms, I'll get bundle monomorphisms), but how do I figure out what the possible rotation classes in that setting are? Actually, I'm not exactly sure how to do it in the Legendrian case either. It seems one has to understand certain homotopy classes of bundle maps, which I don't know anything about. (It's OK if the answer is just a reference to a section of a book on bundle theory or something)

Second, what happens when $\xi$ is not coorientable, that is, there is no global $\alpha$? In this case, I don't even know how to define a complex structure compatible with $\xi$ which seems to be what the rotation class relies on.

This is a follow-up question to my previous one where I was trying to understand the classes of Legendrian immersions of circles into contact manifolds.

I'm interested in classifying isotropic immersions (of spheres, in my particular application, if that makes a difference) into contact manifolds up to regular homotopy through isotropic immersions. Let $(M,\xi)$ be a contact (2n+1)-manifold; here $\xi$ is a distribution of $2n$-planes in $TM$. An isotropic immersion of a manifold N (with dimension $k\leq n$, for $k=n$ this is a Legendrian immersion) is an immersion $f:N\rightarrow M$ where the image of $df_x:T_x N\rightarrow T_{f(x)}M$ lies in the plane $\xi_{f(x)}$ for all $x$.

Mike Usher's answer to my previous question directed me to a paper by Ekholm, Etnyre and Sullivan, in particular section 3.3 (p.19), which defines the rotation class of a Legendrian $f$ as follows.

If $\alpha$ is a contact 1-form on M, that is Ker $\alpha=\xi$, then $d\alpha_p|\xi_p$ is a symplectic form on $\xi_p$. If one chooses any complex structure $J$ for $\xi$ that is compatible with the symplectic structure, then complexifying $df$ to be $df_{\mathbb C}:TL\otimes \mathbb{C}\rightarrow\xi$ is a fiberwise bundle isomorphism.

The rotation class is the homotopy class of $(f,df_{\mathbb C})$ in the space of complex fiberwise isomorphisms $TL\otimes\mathbb{C}\rightarrow \xi$ and is denoted $r(f)$. An h-principle for Legendrian immersions implies that $r(f)$ is a complete invariant for $f$ up to regular homotopy through Legendrian immersions.

First, I can see how the above definition generalizes to isotropic immersions (namely, instead of bundle isomorphisms, I'll get bundle monomorphisms), but how do I figure out what the possible rotation classes in that setting are? Actually, I'm not exactly sure how to do it in the Legendrian case either. It seems one has to understand certain homotopy classes of bundle maps, which I don't know anything about. (It's OK if the answer is just a reference to a section of a book on bundle theory or something)

Second, what happens when $\xi$ is not coorientable, that is, there is no global $\alpha$? In this case, I don't even know how to define a complex structure compatible with $\xi$ which seems to be what the rotation class relies on.