Viability property on Riemannian manifolds

This Note studies a sufficient and necessary condition for the viability property of a state system in a closed subset K of a finite-dimensional compact Riemannian manifold without boundary. Our result is: the system enjoys the viability property in K if and only if the square of the distance function of K is a viscosity supersolution of a second-order partial differential equation in some neighborhood of K. To cite this article: S. Peng, X. Zhu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Brownian bridges on Riemannian manifolds

We study properties of Brownian bridges on a complete Riemannian manifoldM. LetQ _x,y ^t be the law of Brownian bridge fromx toy with lifetimet. Q _x,y ^t is a probability measure on the space _x,y of continuous paths with (0)=x and (1)=y. We prove thatQ _x,y ^t possesses the large deviation property with the rate function

Reflection groups on Riemannian manifolds

We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the orbit space M/G is isometric to a Weyl chamber C which is a Riemannian manifold with corners and certain angle conditions along intersections of faces. We can also reconstruct the manifold and its action from the Riemannian chamber and its equipment of isotropy group data along the faces. We also discuss these results from the point of view of Riemannian orbifolds. Mathematics Subject Classification Primary 51F15, 53C20, 20F55, 22E40     

Clifford structures on Riemannian manifolds

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r=2 and r=3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r=2,3 and 4 respectively, several classes of 8-dimensional manifolds (for 5?r?8), families of real, complex and quaternionic Grassmannians (for r=8,6 and 5 respectively), and Rosenfeld?s elliptic projective planes OP², (C⊗O)P², (H⊗O)P² and (O⊗O)P², which are symmetric spaces associated to the exceptional simple Lie groups F₄, E₆, E₇ and E₈ (for r=9,10,12 and 16 respectively). As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.     

Knots in Riemannian manifolds

In this paper we study submanifolds with nonpositive extrinsic curvature in a positively curved manifold. Among other things we prove that, if ${K\subset (S^n, g)}$ is a totally geodesic submanifold of codimension 2 in a Riemannian sphere with positive sectional curvature where n ≥ 5, then K is homeomorphic to S ^n–2 and the fundamental group of the knot complement ${\pi _1(S^n-K)\cong \mathbb{Z}}$ .     

Av-operator on complete foliated Riemannian manifolds

We give a generalization of the result obtained by C. Currás-Bosch. We consider the Av-operator associated to a transverse Killing fieldν on a complete foliated Riemannian manifold (M, ℱ, g). Under a certain assumption, we prove that, for eachx ∈M, (Av) _x belongs to the Lie algebra of the linear holonomy group ψv(x). A special case of our result, the version of the foliation by points, implies the results given by B. Kostant (compact case) and C. Currás-Bosch (non-compact case).     

Pullback of measures on Riemannian manifolds

A general formula for the pullback of measures on submanifolds of Riemannian manifolds is derived. We prove the equivalence of three different approaches to distributions supported on submanifolds, originating from Gelfand and Shilov, Friedlander, and a characterization using single-layer distributions, respectively.     

Log-Sobolev inequality on non-convex Riemannian manifolds

Counterexamples are constructed to show that when the second fundamental form of the boundary is bounded below by a negative constant, any curvature lower bound is not enough to imply the log-Sobolev inequality. This indicates that in the study of functional inequalities on non-convex manifolds, the concavity of the boundary cannot be compensated by the positivity of the curvature. Next, when the boundary is merely concave on a bounded domain, a criterion on the log-Sobolev inequality known for convex manifolds is proved. Finally, when the concave part of the boundary is unbounded, a Sobolev inequality for a weighted volume measure is established, which implies an explicit sufficient condition for the log-Sobolev inequality to hold on non-convex manifolds.     

Estimates for eigenvalues on Riemannian manifolds

In this paper, we investigate eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain Ω in an n-dimensional complete Riemannian manifold M. When M is an n-dimensional Euclidean space Rn, the conjecture of Pólya is well known: the kth eigenvalue λk of the Dirichlet eigenvalue problem of Laplacian satisfies

     

Sobolev type inequalities on Riemannian manifolds

This paper studies Sobolev type inequalities on Riemannian manifolds. We show that on a complete non-compact Riemannian manifold the constant in the Gagliardo-Nirenberg inequality cannot be smaller than the optimal one on the Euclidean space of the same dimension. We also show that a complete non-compact manifold with asymptotically non-negative Ricci curvature admitting some Gagliardo-Nirenberg inequality is not very far from the Euclidean space.     

Pure pursuit navigation on Riemannian manifolds

Here, the Pure Pursuit navigation on Riemannian manifolds is studied and the time optimal trajectories of this process are found. It has been shown that these trajectories are geodesics of a non-Randers type Finsler metric, known in the literature as Matsumoto metric. A geometrical approach to kinematics in control theory is illustrated and the inverse problem is formulated.     

Levi-parallel contact Riemannian manifolds

We study the Riemannian geometry of contact manifolds with respect to a fixed admissible metric, making the Reeb vector field unitary and orthogonal to the contact distribution, under the assumption that the Levi–Tanaka form is parallel with respect to a canonical connection with torsion.