The index of certain differential operator on tori

Question

Assume that $J$ is an almost complex structure on torus $\mathbb{T}^2$. Let $X$ be a non vanishing vector field on the torus. Let $g$ be a Riemannian metric with corresponding $LC$ connection $\nabla$.

We define the following differential operator on $\chi^{\infty}(\mathbb{T}^2)$, the space of smooth vector fields on tori.

$$ D(Y)=( \nabla_X .\nabla_X +\nabla _{JX}. \nabla_{JX})(Y)$$

Is $D$ an elliptic operator? Does its index depend on $J,X$ and $g$?

When $\nabla, g, J$ are the standard structure on tori, is $D$ the standard Laplacian operator on $\Omega^1(\mathbb{T}^2)\simeq \chi^{\infty}(\mathbb{T}^2)$?(The later equivalency is done via metric $g$).

Answer 1

$D$ is an elliptic operator, as its symbol is given by $$\sigma(D)(p,\xi)=(\xi(X_p))^2+(\xi(JX_p))^2$$ which is non-vanishing for $0\neq\xi\in T_pT^2.$ The index is always 0, as your operator can be deformed continuously (via a family of second order elliptic operators) to the standard Laplacian (acting on pairs of functions). To prove this claim, take a smooth deformation of $J$ and $g$ to the standard $J_0$ and $g_0$ on $T^2$, as well as a scaling of $X$ to a vectorfield $X_0$ of constant length $1$ with respect to $g_0$. At that point, your operator has the same symbol as the standard Laplacian, and you can further deform by scaling the lower order terms to 0 to obtain the standard Laplacian.