Cohomology on a Riemannian foliated manifold with coefficients in the sheaf of germs of foliated currents

✍ Scribed by Mircea Craioveanu; Mircea Puta

• Publisher: John Wiley and Sons
• Year: 1980
• Tongue: English
• Weight: 481 KB
• Volume: 99
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19800990106

✦ Synopsis

Cohomology on a Riemannian foliated manifold with coefficients in the sheaf of germs of foliated currents By MIRCEA CKAIOVEASI; and MIRCEA PUTA of Timipara (Eingegangen am 23. 4. 1979) Summary. Foliated differential f o r m were introduced in [7], [9], to study the cohomology on a RIEMANNian foliated manifold with coefficients in the sheaf of germs of foliated differential forms. In this paper the notion of DE RHAM like current of the type ( p , q)

is defined for a RrEbum-ian foliated manifold and sonic properties of various differential operators acting on the spaces of currents are given. In particular, special DE REAM like currents are considered namely the foliated ones. It turns out that the spam of foliated p-forms is dense in the space of foliated p-currents with the usrial topology. We get certain results concerning the cohomology on a RrEurwian foliated iiitrnifold with coefficients in the sheaf of germs of foliated currents.

Riemannian foliated manifolds

Let P , , , be a n + m-dimensional paraconipact differentiable manifold. We shall consider that differentiable means G". Let E ( V,,,) be the space of differential forms of degree p on Vn+,,,, i.e. the differentiable cross sections of A T*( V,,,,,,).

iiiet.ric (qi,). Then on E ( Vn+,J acts the HODGE operator: P P Next we shall consider that V,,, is a RIEMANNian manifold with the RIEMANNian P defined in the following manner: if ai,,,.iD are the components of LY in a local chart, then: where qi,...ipi,...jm+n-p are the coniponents of the canonical volume element of the metric of P,,, in the same chart. This volume form is determined by: 71 ... n+m = 6 ** = ( -l ) P ( " + m -P ) a , B = det (sir) [21.