Let $({M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation $$\Delta f=S$$

where $\Delta:C^{\infty}(\mathcal{M})\to C^{\infty}({M})$ denotes the Laplace-Beltrami operator, has a solution on $C^{\infty}({M})$ with some source $S\in C^{\infty}({M})$ if and only if

$$\int_{{M}}\,S\,\mathrm{vol}_{g}=0.$$

Furthermore, this solution is unique up to adding a constant, since the only harmonic functions on a compact manifolds are the constant functions.

Now, my question is, is there any literature about the more general case of a Poisson equation acting on tensor fields? More explicitely, under which conditions does the equation

$$\Delta T=S$$

on $\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})$ for some source $S\in\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})$ have a solution, where $\Delta=\mathrm{tr}_{g}(\nabla^{2})$ in this case denotes the connection Laplacian.

Let $({M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation $$\Delta f=S$$

where $\Delta:C^{\infty}({M})\to C^{\infty}({M})$ denotes the Laplace-Beltrami operator, has a solution on $C^{\infty}({M})$ with some source $S\in C^{\infty}({M})$ if and only if

$$\int_{{M}}S\;\mathrm{vol}_{g}=0.$$

Furthermore, this solution is unique up to adding a constant, since the only harmonic functions on a compact manifolds are the constant functions.

Now, my question is, is there any literature about the more general case of a Poisson equation acting on tensor fields? More explicitely, under which conditions does the equation

$$\Delta T=S$$

on $\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})$ for some source $S\in\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})$ have a solution, where $\Delta=\mathrm{tr}_{g}(\nabla^{2})$ in this case denotes the connection Laplacian.

Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation $$\Delta f=S$$

where $\Delta:C^{\infty}(\mathcal{M})\to C^{\infty}(\mathcal{M})$ denotes the Laplace-Beltrami operator, has a solution on $C^{\infty}(\mathcal{M})$ with some source $S\in C^{\infty}(\mathcal{M})$ if and only if

$$\int_{\mathcal{M}}\,S\,\mathrm{vol}_{g}=0.$$

Furthermore, this solution is unique up to adding a constant, since the only harmonic functions on a compact manifolds are the constant functions.

Now, my question is, is there any literature about the more general case of a Poisson equation acting on tensor fields? More explicitely, under which conditions does the equation

$$\Delta T=S$$

on $\Gamma^{\infty}(T^{\ast}\mathcal{M}^{\otimes k})$ for some source $S\in\Gamma^{\infty}(T^{\ast}\mathcal{M}^{\otimes k})$ have a solution, where $\Delta=\mathrm{tr}_{g}(\nabla^{2})$ in this case denotes the connection Laplacian.

Let $({M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation $$\Delta f=S$$

where $\Delta:C^{\infty}(\mathcal{M})\to C^{\infty}({M})$ denotes the Laplace-Beltrami operator, has a solution on $C^{\infty}({M})$ with some source $S\in C^{\infty}({M})$ if and only if

$$\int_{{M}}\,S\,\mathrm{vol}_{g}=0.$$

Furthermore, this solution is unique up to adding a constant, since the only harmonic functions on a compact manifolds are the constant functions.

Now, my question is, is there any literature about the more general case of a Poisson equation acting on tensor fields? More explicitely, under which conditions does the equation

$$\Delta T=S$$

on $\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})$ for some source $S\in\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})$ have a solution, where $\Delta=\mathrm{tr}_{g}(\nabla^{2})$ in this case denotes the connection Laplacian.