The Bochner Laplacian is defined differently from the connection Laplacian, but the two will turn out to differ only by a sign, whenever the former is defined. Let M be a compact, oriented manifold equipped with a metric. Let E be a vector bundle over M equipped a fiber metric and a compatible connection, . This connection gives rise to a differential operator
where denotes smooth sections of E, and T*M is the cotangent bundle of M. It is possible to take the -adjoint of, giving a differential operator
The Bochner Laplacian is given by
which is a second order operator acting on sections of the vector bundle E. Note that the connection Laplacian and Bochner Laplacian differ only by a sign:
Read more about this topic: Laplace Operators In Differential Geometry