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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)
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.)
Full Text: DOI EuDML
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