Let $M$ be a Riemannian manifold, and let $\nabla$ denote its Levi-Civita connection. We have two second order differential operators $\Gamma(T^*M \otimes TM) \to \Gamma(T^*M \otimes TM)$:

The Bochner Laplacian $\Delta_B=\nabla^* \nabla$, and the "Hodge" Laplacian $\Delta_H=\delta_{\nabla} d_{\nabla}+d_{\nabla}\delta_{\nabla}$, where here $d_{\nabla}: \Omega^1(M,TM) \to \Omega^2(M,TM)$ is the covariant exterior derivative associated with the Levi-Civita connection $\nabla$, and $\delta_{\nabla}$ is its adjoint.

Is there some kind of Weitzenböck formula connecting these two Laplacians? something like $\Delta_H-\Delta_B=C(R^{\nabla})$, where $R^{\nabla}$ is the curvature tensor of $\nabla$?

I know that for real-valued one-forms, the classical Weitzenböck formula is $\Delta_H-\Delta_B=\text{Ric}$, where "$\text{Ric}$" denotes the Ricci curvature of the manifold. Can we do something similar here?

  • $\begingroup$ Please look at this question and the discussion in the comments there. In particular, please pay attention to P.Peterson's note where this topic is covered in a greater generality. $\endgroup$ – Yuri Vyatkin Apr 20 at 23:58
  • $\begingroup$ A proof of a similar identity for the case of vector-bundle-valued 1-forms one can find on p.18 of my thesis. Perhaps, specializing that calculation for the case of $TM$-valued 1-forms would be a partial answer to your question. $\endgroup$ – Yuri Vyatkin Apr 21 at 0:30
  • $\begingroup$ Thank you, I will take a look. $\endgroup$ – Asaf Shachar Apr 21 at 14:41

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