Connection Laplacian

Let $E\to M$ be a smooth vector bundle with connection $\nabla^E$ , and let $g$ be a pseudo-Riemannian metric with Levi–Civita connection $\nabla^{\mathrm{LC}}$ . The second covariant derivative of a section $s$ is

(\nabla^E)^2_{X,Y}s =\nabla^E_X\nabla^E_Ys-\nabla^E_{\nabla^{\mathrm{LC}}_X Y}s.

In the sign convention of this collection, the connection Laplacian is

\Delta_{\nabla^E}s=-\operatorname{tr}_g\bigl((\nabla^E)^2s\bigr).

In a local pseudo-orthonormal frame $e_i$ , with $g(e_i,e_i)=\varepsilon_i\in\{+1,-1\}$ , this is

\Delta_{\nabla^E}s=-\sum_i\varepsilon_i\Bigl(\nabla^E_{e_i}\nabla^E_{e_i}s-\nabla^E_{\nabla^{\mathrm{LC}}_{e_i}e_i}s\Bigr).

Special cases

For a Riemannian metric and a metric connection, this is also called the rough or Bochner Laplacian and has nonnegative leading sign. For a Lorentzian metric it is a connection wave operator and is normally hyperbolic. On the trivial line bundle with its trivial connection, it reduces to the scalar Laplace–Beltrami operator.

References

Peter Petersen, Riemannian Geometry, 3rd ed., Springer, 2016. Publisher record. Relevant: Chapter 4.
Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: §1.5.