— In this article we will give a brief summary of some recent work on two variational problems in Riemannian geometry. Although both involve the study of elementary symmetric functions of the eigenvalues of the Ricci tensor, as far as technique and motivation are concerned the problems are actually quite different. Résumé (Problèmes variationnels locaux et non-locaux en géométrie riemannienne) ...

We establish necessary and sufficient conditions for existence of isometric immersions of a simply connected Riemannian manifold into a two-step nilpotent Lie group. This comprises the case of immersions into H-type groups. MSC 2000: 53C42, 53C30

We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces. Recently, Oshima [67] published his formula for c-function for L 2 on pseudo-Riemannian symmetric spaces (see also works of Delorm [12] and van den Ban– Schlichtkrull [3], [4]). After this, there arises a natural question about other solvable problems of non-commutative harmonic analysis. In the Appendix to the paper...

Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We derive geodesic equations and a formula for sectional curvature for conformally equivalent metrics. We show if the conformal factor depends only on the length of the curve, the metric behaves like an L metric, the sectional curvature is not bounded f...

and Applied Analysis 3 in the normal bundle T⊥M, and AN is the shape operator of the second fundamental form. Moreover, we have g ANX, Y g h X,Y ,N , 2.4 where g denotes the Riemannian metric onM as well as the metric induced onM. The mean curvature vector H on M is given by

Let L be a second order elliptic differential operator on a Riemannian manifold E with no zero order terms. We say that a function h is L-harmonic if Lh = 0. Every positive L-harmonic function has a unique representation

Let M a Riemannian manifold of dimension m ≥ 3, let Σ be a closed smooth submanifold of M of co-dimension at least 2, and let H be a Hadamard manifold with pinched sectional curvatures. We prove the existence and uniqueness of harmonic maps φ : M \ Σ → H with prescribed singularities along Σ. When M = R, and H = H C , the complex hyperbolic space, this result has applications to the problem of ...

X iv :0 90 5. 13 66 v2 [ m at h. SP ] 2 1 M ay 2 00 9 Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary Yiqian Shi ∗† and Bin Xu∗⋆ Abstract. Let eλ(x) be an eigenfun tion with respe t to the Lapla e-Beltrami operator ∆M on a ompa t Riemannian manifoldM without boundary: ∆Meλ = λ2eλ. We show the following gradient estimate of eλ: for every λ≥ 1, there holds ...

We consider a Hamiltonian setup (M, ω,H,L,Γ,P), where (M, ω) is a symplectic manifold, L is a distribution of Lagrangian subspaces in M, P a Lagrangian submanifold of M, H is a smooth time dependent Hamiltonian function on M and Γ : [a, b] 7→ M is an integral curve of the Hamiltonian flow ~ H starting at P . We do not require any convexity property of the Hamiltonian function H . Under the assu...