Let $M$ be an oriented Riemannian manifold. In this lecture we make contact with some of the differential operators you first met in Analysis II:

  • The divergence $ \operatorname{div}(X)$ of a vector field,
  • The gradient $ \operatorname{grad}(f)$ of a smooth function,
  • The Hessian $ \operatorname{Hess ^{ \nabla}}(f)$ of a smooth function,
  • The Laplacian $ \Delta(f)$ of a smooth function.

But why define something once when you can define it three times? 🙃

For the sake of completeness we give three equivalent definitions of the Laplacian $ \Delta(f)$:

  1. $ \Delta(f) = \operatorname{div}( \operatorname{grad}(f))$,
  2. $ \Delta(f) = - \delta(df)$,
  3. $ \Delta(f) = \operatorname{tr}( \operatorname{Hess}^{ \nabla}(f))$.

Comments and questions?