We prove an integral formula on closed oriented manifolds equipped with a codimension two foliation whose leaves are compact.

We prove an analogue of mean curvature comparison theorem in the case where the Ricci curvature below a positive constant is small in $L^1$-norm.

Recently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of $H$-harmonic and $H^*$-harmonic spaces associated to a H\"ormander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.

This paper studies the following question: Given an $\omega'$-symplectic action of a Lie group on a manifold $M$ which coincides, as a smooth action, with a Hamiltonian $\omega$-action, when is this action a Hamiltonian $\omega'$-action? Using a result of Morse-Bott theory presented in Section~2, we show in Section~3 of this paper that such an action is in fact a Hamiltonian $\omega'$-action, provided that $M$ is compact and that the Lie group is compact and connected. This result was first proved by Lalonde-McDuff-Polterovich in 1999 as a consequence of a more general theory that made use of hard geometric analysis. In this paper, we prove it using classical methods only.

In this paper we show that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect. This is then generalized to show that in manifolds with positive Ricci curvature in the integral sense two minimal hypersurfaces must be close to each other. We also show what happens if a manifold with nonnegative Ricci curvature admits two nonintersecting minimal hypersurfaces.

A partial differential equation, the local M\"obius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local M\"obius equation and an additional partial differential equation.

Si $M$ est une vari\'et\'e de dimension $n$, compacte non simplement connexe, on caract\'erise les m\'etriques riemanniennes sur $M$ dont la fonction croissance a exactement deux singularit\'es.

The geometry of indicatrices is the foundation of Minkowski geometry. A strongly convex indicatrix in a vector space is a strongly convex hypersurface. It admits a Riemannian metric and has a distinguished invariant---(Cartan) torsion. We prove the existence of non-trivial strongly convex indicatrices with vanishing mean torsion and discuss the relationship between the mean torsion and the Riemannian curvature tensor for indicatrices of Randers type.

We classify finite subgroups of $\SO(4)$ generated by anti-unitary involutions. They correspond to involutions fixing pointwise a Lagrangian plane. Explicit descriptions of the finite groups and the configurations of Lagrangian planes are obtained.

In this paper we prove that there is no nontrivial $L^{q}$-integrably $p$-harmonic $1$-form on a complete manifold with nonnegatively Ricci curvature $(0
Keywords:$p$-harmonic, $1$-form, complete manifold, Sobolev inequality
Categories:58E20, 53C21

In this paper the germ of neighborhood of a compact leaf in a Lagrangian foliation is symplectically classified when the compact leaf is $\bT^2$, the affine structure induced by the Lagrangian foliation on the leaf is complete, and the holonomy of $\bT^2$ in the foliation linearizes. The germ of neighborhood is classified by a function, depending on one transverse coordinate, this function is related to the affine structure of the nearly compact leaves.

Motivated by deformation theory of holomorphic maps between almost complex manifolds we endow, in a natural way, the tangent bundle of an almost complex manifold with an almost complex structure. We describe various properties of this structure.

The Hamiltonian potentials of the bending deformations of $n$-gons in $\E^3$ studied in \cite{KM} and \cite{Kl} give rise to a Hamiltonian action of the Malcev Lie algebra $\p_n$ of the pure braid group $P_n$ on the moduli space $M_r$ of $n$-gon linkages with the side-lengths $r= (r_1,\dots, r_n)$ in $\E^3$. If $e\in M_r$ is a singular point we may linearize the vector fields in $\p_n$ at $e$. This linearization yields a flat connection $\nabla$ on the space $\C^n_*$ of $n$ distinct points on $\C$. We show that the monodromy of $\nabla$ is the dual of a quotient of a specialized reduced Gassner representation.

Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in $\R^3$ that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the B\"acklund transformation for constant torsion curves in $\R^3$, previously derived from the well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in $H^3$ or $S^3$ leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.

A new class of 3-dimensional contact metric manifolds is found. Moreover it is proved that there are no such manifolds in dimensions greater than 3.

We obtain a homotopy splitting of the forget control map for controlled homeomorphisms over closed manifolds of nonpositive curvature.

We shall provide a characterization of all isoparametric hypersurfaces $M$'s in a real space form $\tilde{M}(c)$ by observing the extrinsic Wshape of geodesics of $M$ in the ambient manifold $\tilde{M}(c)$.

An integral formula is derived for the spherical functions on the symmetric space $G/K=\break \SO_0(p,q)/\SO(p)\times \SO(q)$. This formula allows us to state some results about the analytic continuation of the spherical functions to a tubular neighbourhood of the subalgebra $\a$ of the abelian part in the decomposition $G=KAK$. The corresponding result is then obtained for the heat kernel of the symmetric space $\SO_0(p,q)/\SO (p)\times\SO (q)$ using the Plancherel formula. In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel.

Bessa [Be] proved that for given $n$ and $i_0$, there exists an $\varepsilon(n,i_0)>0$ depending on $n,i_0$ such that if $M$ admits a metric $g$ satisfying $\Ric_{(M,g)} \ge n-1$, $\inj_{(M,g)} \ge i_0>0$ and $\diam_{(M,g)} \ge \pi-\varepsilon$, then $M$ is diffeomorphic to the standard sphere. In this note, we improve this result by replacing a lower bound on the injectivity radius with a lower bound of the conjugate radius.

We study real hypersurfaces of a complex space form $M_n(c)$, $c\ne 0$ under certain conditions of the Ricci tensor on the orthogonal distribution $T_o$.

In this paper, we define the $\eta$-invariant for a cusped hyperbolic $3$-manifold and discuss some of its applications. Such an invariant detects the chirality of a hyperbolic knot or link and can be used to distinguish many links with homeomorphic complements.

Given a continuous map $\delta$ from the circle $S$ to itself we want to find all self-maps $\sigma\colon S\to S$ for which $\delta\circ\sigma = \delta$. If the degree $r$ of $\delta$ is not zero, the transformations $\sigma$ form a subgroup of the cyclic group $C_r$. If $r=0$, all such invertible transformations form a group isomorphic either to a cyclic group $C_n$ or to a dihedral group $D_n$ depending on whether all such transformations are orientation preserving or not. Applied to the tangent image of planar closed curves, this generalizes a result of Bisztriczky and Rival [1]. The proof rests on the theorem: {\it Let $\Delta\colon\bbd R\to\bbd R$ be continuous, nowhere constant, and $\lim_{x\to -\infty}\Delta(x)=-\infty$, $ \lim_{x\to+\infty}\Delta (x)=+\infty$; then the only continuous map $\Sigma\colon\bbd R\to\bbd R$ such that $\Delta\circ\Sigma=\Delta$ is the identity $\Sigma=\id_{\bbd R}$.