# The heat kernel in Hermitian bundles over Riemannian manifolds

In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold.

Could you please point me to texts doing the same kind of construction (not necessarily using densities) in vector bundles over arbitrary (i.e. not necessarily compact anymore) Riemannian manifolds for the connection Laplacian?

I am mostly interested in the clear definition of the object and the statement of its basic properties. Do the positivity and minimality properties of the heat kernel on functions have analogous properties in bundles?

• The construction of heat kernel is something local since Laplacian is defined only locally. Thus the construction you learned can be extended to the interior of any manifolds. For the construction of heat kernel over manifold with boundary, there are many references available. One thing you may clarify is that you probably need some boundary condition imposed on the metric. Otherwise the Laplacian on the buondary is technically no different from the Laplacian on a different submanifold. – Bombyx mori Sep 13 at 21:23
• @Bombyxmori: Thank you, but I was looking for a reference, i.e. a text containing all the details from bottom to top. Should I understand your comment as suggesting that such a text might not exist? – Alex M. Sep 14 at 13:22
• M: There are plenty of such books. One book you can use is Yau and Schoen's book "Lecture on Differential Geometry". A lot of it is on construction of the heat kernel. However I never read the book myself. You may also try Richard Melrose' book "Atiyah Singer Patodi index theorem". Both are very dense and can be hard to read. – Bombyx mori Sep 14 at 17:30
• @Bombyxmori: Melrose's construction is for bundles over compact manifolds, then extended to copmpact "b"-manifolds. Yau's and Schoen's construction is for the trivial bundle $M \times \mathbb R$ over a complete manifold with non-zero injectivity radius. None of these is what I want. – Alex M. Sep 21 at 6:50
• This is actually not entirely true. Melrose's construction can be extended to even worse situation than manifold with boundary, like cusp singularity. I never read Yau and Schoen's book in detail, but I am kind of skeptical if their method is as restrictive as you said. – Bombyx mori Sep 21 at 7:04