Iterated Differential Forms II: Riemannian Geometry Revisited

Separation conditions for iterated function systems with overlaps on Riemannian manifolds

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau \textit{et al.} in [Monatsch. Math. 156 (2009), 325-355]. We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin-Yau [Comm. Anal. Geom. 13 (2005), 821--843] and Lau-Ngai [Adv. Math. 208 (2007), 647-671] on Euclidean spaces. Moreover, we obtain a for…

Uniqueness Results on a geometric PDE in Riemannian and CR Geoemetry Revisited

We revisit some uniqueness results for a geometric nonlinear PDE related to the scalar curvature in Riemannian geometry and CR geometry. In the Riemannian case we give a new proof of the uniqueness result assuming only a positive lower bound for Ricci curvature. We apply the same principle in the CR case and reconstruct the Jerison-Lee identity in a more general setting. As a consequence we prove a stronger uniqueness result in the CR case. We also discuss some open problems for further study.

Lovelock's theorem revisited

Let (X, g) be an arbitrary pseudo-riemannian manifold. A celebrated result by Lovelock gives an explicit description of all second-order natural (0,2)-tensors on X, that satisfy the conditions of being symmetric and divergence-free. Apart from the dual metric, the Einstein tensor of g is the simplest example. In this paper, we give a short and self-contained proof of this theorem, simplifying the existing one by formalizing the notion of derivative of a natural tensor.

Compact Clifford-Klein forms of symmetric spaces -- revisited

This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space $G/H$ if $H$ is non-compact. The first half of the article elucidates general machinery to study discontinuous groups for $G/H$, followed by the most update and complete list of symmetric spaces with/without compact quotients. In the second half, as applications of general theory, we prove: (i) there exists a 15 dimensional compact pseudo-Riemannian manifold of sign…

黎曼几何中的喷射同构定理

The Jet Isomorphism Theorem of Riemannian Geometry

本文通过坐标无关方法，利用杨氏对称化器的内在定义，实现了曲率张量喷射的重建，解决了黎曼几何中喷射同构定理的问题。

从扩散中恢复黎曼几何

Recovering Riemannian Geometry from Diffusion

本文提出通过扩散半群内在重建黎曼几何的方法，揭示几何结构由扩散行为决定，无需预设度量或坐标。