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The Bochner Laplacian, Riemannian submersions, heat content asymptotics, and heat equation asymptotics. (English) Zbl 0955.53022
Let $$\pi : Z \rightarrow Y$$ be a Riemannian submersion where the fibers of $$\pi$$ are compact and where $$Y$$ is a compact manifold with smooth boundary. The main result of the paper reads as follows. Let $$\Delta ^0_Y$$ or $$\Delta ^0_Z$$ be the usual scalar Laplacian of $$Y$$ or of $$Z$$, respectively. Then $$\pi ^*\Delta ^0_Y = \Delta ^0_Z \pi ^*$$ if and only if the fibers of $$\pi$$ are minimal submanifolds of $$Z$$. Moreover, some formulae for the heat kernel of the corresponding Bochner Laplacians are deduced.
Reviewer: I.Kolář (Brno)
##### MSC:
 53C20 Global Riemannian geometry, including pinching 58J35 Heat and other parabolic equation methods for PDEs on manifolds 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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