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Heat conduction for Riemannian foliations

Seiki Nishikawa, Mohan Ramachandran, Philippe Tondeur

1989
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Bulletin of the American Mathematical Society
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A foliation Jona manifold M is a partition of M into submanifolds, the leaves of M, which locally looks like a family of parallel subspaces in Euclidean space. This gives rise to two types of geometries, namely tangential and transversal. More specifically let gM be a Riemannian metric on M. It induces on each leaf a Riemannian metric, and hence a corresponding leafwise Laplacian A 0 . The action of the corresponding semigroup e~t A° on the bigraded de Rham complex Q M is studied in [AT], and
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... died in [AT], and leads to a tangential or leafwise Hodge decomposition theorem. A foliation & is Riemannian [R], if the induced Riemannian metric gQ on the normal bundle Q = TM\L y L the tangent bundle of ^, is holonomy invariant, i.e. 0(X)gQ = 0 for all vector fields X tangent to &. This gives rise to a transversal Riemannian geometry, which can heuristically be thought of as the Riemannian geometry of the (singular) space of leaves B = M/P'. The complex Çi B {^) c QM of forms co satisfying i(X)co = 0 (interior product) and 9(X)co = 0 (Lie derivative) for all X e TL is the complex of basic differential forms of £F, and heuristically plays the role of the de Rham complex of the leaf space B. The transversal Riemannian metric gQ gives rise to a transversal or basic Laplacian A#: QB{^) -• QB^)- The main point of this announcement is to construct and study the corresponding semigroup e~t As acting on Q^(y), and to examine its limit behavior for t -• oo. This yields in particular a new proof of the Hodge decomposition theorem in Q B (^)- The Laplacian A B = dsS B + SBCIB is formally constructed from the transversal Riemannian geometry in the normal bundle in the usual fashion. Since the basic differential forms do not constitute all sections of a vector bundle, the usual elliptic theory does not apply directly. A technical device to handle this situation is to extend A B : SIB{SF) -> 0^B{^) to a genuine elliptic operator A: fi(M) -• ii(M). An explicit construction of such an extension was given in [KT]. This involves the assumption that the mean curvature is constant along the leaves. This hypothesis enters in the explicit calculation of the formal adjoint SB of ds, and hence also of A B . It is further used in the construction of the extension A. Formally this hypothesis is expressed by dualizing the usual mean curvature vector field to a 1-form K (vanishing along the leaves of ^), and requiring it to be an element of Çï x B {^).

doi:10.1090/s0273-0979-1989-15826-6
fatcat:d3fpjln34bbb7l7gqbkttbebvu