We prove that higher moment maps on area measures of a euclidean vector space are injective, while the kernel of the centroid map equals the image of the first variation map. Based on this, we introduce the space of smooth dual area measures on a finite-dimensional euclidean vector space and prove that it admits a natural convolution product which encodes the local additive kinematic formulas for groups acting transitively on the unit sphere. As an application of this new integral-geometric structure, we obtain the local additive kinematic formulas in hermitian vector spaces in a very explicit way.
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Alesker, S.: The multiplicative structure on continuous polynomial valuations. Geom. Funct. Anal. 14(1), 1–26 (2004)
Bernig, A.: Algebraic integral geometry. In: Global Differential Geometry. Volume 17 of Springer Proceedings in Mathematics, pp. 107–145. Springer, Berlin (2012)
Bernig, A.: Invariant valuations on quaternionic vector spaces. J. Inst. Math. Jussieu 11, 467–499 (2012)
Bernig, A., Bröcker, L.: Valuations on manifolds and Rumin cohomology. J. Differ. Geom. 75(3), 433–457 (2007)
Bernig, A., Faifman, D.: Generalized translation invariant valuations and the polytope algebra. Adv. Math. 290, 36–72 (2016)
Bernig, A., Fu, J.H.G.: Convolution of convex valuations. Geom. Dedicata 123, 153–169 (2006)
Bernig, A., Fu, J.H.G.: Hermitian integral geometry. Ann. Math. 173, 907–945 (2011)
Bernig, A., Fu, J.H.G., Solanes, G.: Integral geometry of complex space forms. Geom. Funct. Anal 24(2), 403–492 (2014)
Bernig, A., Fu, J.H. G., Solanes, G.: Dual curvature measures in Hermitian integral geometry. In: Analytic Aspects of Convexity, volume 25 of Springer INdAM series, pp. 1–17. Springer, Cham (2018)
Bernig, A., Hug, D.: Kinematic formulas for tensor valuations. J. Reine Angew. Math. 736, 141–191 (2018)
Fu, J.H.G.: Kinematic formulas in integral geometry. Indiana Univ. Math. J. 39(4), 1115–1154 (1990)
Fu, J.H.G.: Structure of the unitary valuation algebra. J. Differ. Geom. 72(3), 509–533 (2006)
Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin (1957)
Huybrechts, D.: Complex Geometry: An Introduction. Springer, Berlin (2005)
Klain, D.A., Rota, G.-C.: Introduction to Geometric Probability. Lezioni Lincee. [Lincei Lectures]. Cambridge University Press, Cambridge (1997)
McMullen, P.: Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc. (3) 35(1), 113–135 (1977)
Nijenhuis, A.: On Chern’s kinematic formula in integral geometry. J. Differ. Geom. 9, 475–482 (1974)
Park, H.: Kinematic formulas for the real subspaces of complex space forms of dimension \(2\) and \(3\). Ph.D. thesis, University of Georgia (2002)
Rumin, M.: Differential forms on contact manifolds. J. Differ. Geom. 39(2), 281–330 (1994)
Solanes, G.: Contact measures in isotropic spaces. Adv. Math. 317, 645–664 (2017)
Wannerer, T.: Integral geometry of unitary area measures. Adv. Math. 263, 1–44 (2014)
Wannerer, T.: The module of unitarily invariant area measures. J. Differ. Geom. 96(1), 141–182 (2014)
I thank Gil Solanes for many useful comments on a first version of this paper.
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Bernig, A. Dual area measures and local additive kinematic formulas. Geom Dedicata 203, 85–110 (2019). https://doi.org/10.1007/s10711-019-00427-3
- Area measure
- Kinematic formula
- Integral geometry
Mathematics Subject Classification (2000)