A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic line

Integral geometry of complex space forms

We show how Alesker's theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this appr…

Hermitian integral geometry

We give in explicit form the principal kinematic formula for the action of the affine unitary group on $\C^n$, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce also several other canonical bases for the algebra of unitary-invariant valuations, explore their interrelations, and characterize in these terms the cones of positive and monotone elements.

Valuations and curvature measures on complex spaces

We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local kinematic formulas on such spaces were recently obtained. While the local and global kinematic formulas in the Euclidean case are formally identical, the local formulas in the hermitian case contain strictly more information than the global ones. Even if one is only interested in the flat hermitian case, i.e. $\mathbb C^n$, it is necessary to study the family of a…

Algebraic integral geometry

A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.

Convolution of convex valuations

We show that the natural "convolution" on the space of smooth, even, translation-invariant convex valuations on a euclidean space $V$, obtained by intertwining the product and the duality transform of S. Alesker, may be expressed in terms of Minkowski sum. Furthermore the resulting product extends naturally to odd valuations as well. Based on this technical result we give an application to integral geometry, generalizing Hadwiger's additive kinematic formula for $SO(V)$ to general compact groups $G \subset O(V)$ acting transitively on the sphere: it turns out that these formulas are in a natur…

Integral geometry of exceptional spheres

The algebras of valuations on $S^6$ and $S^7$ invariant under the actions of $\mathrm G_2$ and $\mathrm{Spin}(7)$ are shown to be isomorphic to the algebra of translation-invariant valuations on the tangent space at a point invariant under the action of the isotropy group. This is in analogy with the cases of real and complex space forms, suggesting the possibility that the same phenomenon holds in all Riemannian isotropic spaces. Based on the description of the algebras the full array of kinematic formulas for invariant valuations and curvature measures in $S^6$ and $S^7$ is computed. A key t…

The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold

The main technical result of the paper is a Bochner type formula for the sub-laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the $Sp(n)Sp(1)$ components of the qc-Ricci curvature. It is shown that in the case of a 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is a round 3-Sasakian sphere. Another goal of the paper is to establish a-priori estimates for square integrals of horizontal derivatives of …

Integral geometry under $G_2$ and $Spin(7)$

A Hadwiger-type theorem for the exceptional Lie groups $G_2$ and $Spin(7)$ is proved. The algebras of $G_2$ or $Spin(7)$ invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formulas for these groups are determined.

四元数接触几何中热方程的Li-Yau次梯度估计与Perelman型熵公式

Li-Yau sub-gradient estimates and Perelman-type entropy formulas for the heat equation in quaternionic contact geometry

本文在紧致四元数接触流形上建立qc热方程的Li-Yau型次梯度估计，并在qc-爱因斯坦流形上推导Perelman型熵公式，推广了经典黎曼几何结果。

Contact integral geometry and the Heisenberg algebra

Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. In this note, we uncover a similar structure for contact manifolds. Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings. Those valuations assign numerical invariants to even-dimensional submanifo…

四重极结构与局部四元数 торic 动作的兼容性

Tetraplectic structures compatible with local quaternionic toric actions

本文引入了四元数的 toric 几何类比，通过发展局部 $Q^n := Sp(1)^n$-作用理论，研究四元数 torus 作用的分类与特征对和欧拉类等不变量。

四元数 LVM 流形

Quaternionic LVM Manifolds

本文提出四元数 LVM 流形（LVMQ）构造，通过四元数矩角积得到紧致商流形，研究其连通性、Pontryagin 类及四元数环面结构，并探讨秩一情形的 Hopf 纤维化。