Journal of Differential Geometry

Convolution of valuations on manifolds

Semyon Alesker and Andreas Bernig

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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.


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


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

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX] 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]


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

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