## Journal of Differential Geometry

### Convolution of valuations on manifolds

#### Abstract

We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and generalized valuations on $M$ are modules over the algebra of compactly supported generalized valuations on $G$ satisfying some technical condition of tameness.

The case of a vector space acting on itself is studied in detail. We prove explicit formulas in this case and show that the new convolution is an extension of the convolution on smooth translation invariant valuations introduced by J. Fu and the second named author.

#### Note

S. A. was partially supported by ISF grant 1447/12.

#### Note

A. B. was supported by DFG grant BE 2484/5-1.

#### Article information

Source
J. Differential Geom., Volume 107, Number 2 (2017), 203-240.

Dates
Received: 5 August 2015
First available in Project Euclid: 29 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1506650420

Digital Object Identifier
doi:10.4310/jdg/1506650420

Mathematical Reviews number (MathSciNet)
MR3707644

Zentralblatt MATH identifier
1372.52018

#### Citation

Alesker, Semyon; Bernig, Andreas. Convolution of valuations on manifolds. J. Differential Geom. 107 (2017), no. 2, 203--240. doi:10.4310/jdg/1506650420. https://projecteuclid.org/euclid.jdg/1506650420