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The Gauss formulas for Laplacians on submanifolds
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abstract
There are several types of Laplacians of a vector field on a Riemannian manifold. These include the Bochner and the Hodge Laplacian. The Gauss formula for the Levi-Civita connection relates the extrinsic connection to the intrinsic connection. We extend the Gauss formula for the connection to formulas for the different types of Laplacians of a vector field on a submanifold of any codimension $k\geq 1$. In the process, we derive a Gauss formula for the Ricci operator, formulas for the divergence of the second fundamental form, and a formula for the Laplacian of a $1$-form on a surface of revolution in terms of the Lie derivatives. The formulas have applications to the study of the formulation of the incompressible Navier-Stokes equations on a Riemannian manifold.
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