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 ...

where r = ∫ Rdμ/ ∫ dμ is the average scalar curvature (R is the scalar curvature) and Ric is the Ricci curvature tensor of h. Hamilton then spectacularly illustrated the success of this method by proving, when n = 3, that if the initial Riemannian metric has strictly positive Ricci curvature it evolves through time to a positively curved Einstein metric h∞ on M . And, because n = 3, such a Riem...

In this paper, we study the problem of facial expression recognition using a novel space-time geometric representation. We describe the temporal evolution of facial landmarks as parametrized trajectories on the Riemannian manifold of positive semidefinite matrices of fixed-rank. Our representation has the advantage to bring naturally a second desirable quantity when comparing shapes – the spati...

and the corresponding Euler-Lagrange equation is H = 0, where H is the mean curvature vector field. If φ : (M, g) → (N, h) is a Riemannian immersion, then it is a critical point of the bienergy in C∞(M,N) if and only if it is a minimal immersion [26]. Thus, in order to study minimal immersions one can look at harmonic Riemannian immersions. A natural generalization of harmonic maps and minimal ...

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 l...

Abstract. Let B1 be the unit open disk in R2 and M be a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in H1([0, T ]×B1,M) whose energy is non-increasing in time, given initial data u0 ∈ H(B1,M) and boundary data γ = u0|∂B1 . Previously, this uniqueness result was obtained by Rivière (when M is the round sphere and the en...

Abstract- Kernel trick and projection to tangent spaces are two choices for linearizing the data points lying on Riemannian manifolds. These approaches are used to provide the prerequisites for applying standard machine learning methods on Riemannian manifolds. Classical kernels implicitly project data to high dimensional feature space without considering the intrinsic geometry of data points. ...

Using techniques both of non linear analysis and geometric measure theory, we prove existence of minimizers and more generally of critical points for the Willmore functional and other Lp curvature functionals for immersions in Riemannian manifolds. More precisely, given a 3-dimensional Riemannian manifold (M, g) and an immersion of a sphere f : S2 ↪→ (M, g) we study the following problems. 1) T...

One of the major tools introduced by Perelman is his reduced volume ̃ V [21, Sect. 7]. This is a certain geometric quantity which is monotonically nondecreasing in time when one has a Ricci flow solution. Perelman’s main use of the reduced volume was to rule out local collapsing in a Ricci flow. Before giving his rigorous proof that ̃ V is monotonic, Perelman gave a heuristic argument [21, Sect. ...

We study a well-known scalar quantity in Riemannian geometry, the Ricci scalar, in the context of diffusion tensor imaging (DTI), which is an emerging non-invasive medical imaging modality. We derive a physical interpretation for the Ricci scalar and explore experimentally its significance in DTI. We also extend the definition of the Ricci scalar to the case of high angular resolution diffusion...