Integral geometry of complex space forms
Integral geometry of complex space forms
复制标题
复杂空间形式的整体几何
DOI：
10.1007/s00039-014-0251-1
复制
发表时间：
2012
影响因子：
2.2
通讯作者：
G. Solanes
中科院分区：
文献类型：
--
作者：
A. Bernig;Joseph H. G. Fu;G. Solanes
我们展示了如何阿列斯克理论的价值流形上产生的代数图片的积分几何的任何黎曼各向同性空间。然后我们应用这种方法对复射影空间、复双曲空间和复欧氏空间等复空间形式的积分几何作了详细的说明。特别是，我们计算家庭的运动公式不变的价值观和不变的曲率措施在这些空间。除了Gray的管公式和Shifrin的运动学公式的新的和更有效的框架，这种方法产生了一个新的公式表示的体积管的一个完全真实的子流形的内在黎曼结构。我们还表明，通过直接计算的Lipschitz-Killing赋值稳定不变的角曲率措施的子空间，这表明一个类似的现象持有所有黎曼流形的可能性。最后，我们提出了一些悬而未决的问题和建议。
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 approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.