In this talk we first recall the notion of a subriemannian manifold and we provide various examples. Under some additional assumptions it is known that a subriemannian structure induces a hypo-elliptic positive operator which is called sub-Laplacian. In the cases where the manifold is a sphere (of a certain dimension) or a general compact two step nilmanifold we study the heat kernel, the spectrum and the spectral zeta function of the sub-Laplacian.

This presentation is based on a joint work with K. Furutani (Tokyo University of Science, Japan) and C. Iwasaki (University of Hyogo, Japan).

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