Tube formulas for valuations in complex space forms (Q6663194)

The paper is about tube formulas in Riemannian manifolds, and in particular in complex space forms. This topic has a rich history. Weyl's famous tube formula [\textit{H. Weyl}, Am. J. Math. 61, 461--472 (1939; Zbl 0021.35503)] has led to a proof of the Chern-Gauss-Bonnet by \textit{C. B. Allendoerfer} and \textit{A. Weil} [Trans. Am. Math. Soc. 53, 101--129 (1943; Zbl 0060.38102)] Later on, Gray studied the volume of tubes in great details, he even wrote a book with the title ``Tubes'' [\textit{A. Gray}, Trubki. Formula Vejlya i ee obobshcheniya (Russian). Moscow: Mir (1993; Zbl 0849.53001)]. The case of real and complex space forms also attracted a lot of attention, for instance by [\textit{L. A. Santalo}, Proc. Am. Math. Soc. 1, 325--330 (1950; Zbl 0041.29301)], \textit{T. Shifrin} [Trans. Am. Math. Soc. 264, 255--293 (1981; Zbl 0473.53058)], \textit{P. A. Griffiths} [Duke Math. J. 45, 427--512 (1978; Zbl 0409.53048)], \textit{A. Gray} and \textit{L. Vanhecke} [Rend. Semin. Mat., Torino 39, No. 3, 1--50 (1981; Zbl 0511.53059)] and \textit{A. Gray} [Trans. Am. Math. Soc. 291, 437--449 (1985; Zbl 0575.53043)]. The paper under review takes up these lines of studies and brings in a new tool, the relatively recent theory of valuations on manifolds developed by Alesker and others. Using this theory allows to look at invariants of tubes other than the volume, and the natural question, which is studied in the paper, is what kind of tube formulas exist in this setting. The authors first give a general formula valable in any Riemannian manifold and then specify it to the case of complex space forms, where they do explicit computations of the whole array of tube formulas.\N\NIn Section 3, the tube formulas in \(\mathbb R^n\) and \(\mathbb C^n\) are worked out. The main point is the existence of an \(\mathrm{sl}_2\)-representation in both cases, which makes it possible to write the tube operators in terms of primitive elements.\N\NIn Section 4, a general framework for tubes in Riemannian manifolds is described using Alesker's theory of valuations. Then the boundary operator is computed for real and complex space forms. Using some isomorphism, the boundary operator can be related to the \(\mathrm{sl}_2\)-representation in the flat case, which is an important observation for the following computations.\N\NIn Section 5, all computations are carried out in an irreducible \(\mathrm{sl}_2\)-representation, where the operator \(Y_\lambda=Y-\lambda X\) is studied in detail, including its spectrum and image. This is then translated into explicit tube formulas on real and complex space forms in Section 6.