A Counter Example To the Hodge Conjecture

In this paper, we give a simple counter example to the famous Hodge conjecture

Complex manifolds with generating tangent bundles

Let $M$ be a close complex manifold and $TM$ its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then $M$ is a complex homogeneous manifold. Our proof depends on the complex version of Chow-Rashevskii theorem in Carnot-Caratheodory spaces

On Existences Of Periodic Orbits For Hamilton Systems

We prove that either there exists at least one hamilton periodic orbit in a given energy close smooth hypersurface or there exist at least two hamilton periodic orbits in a near-by energy close smooth hypersurface. More general results also hold

Simple homotopy invariance of higher signatures

We prove that the higher signature for any close oriented manifold is a simple-homotopy invariant

A Proof On Weinstein Conjecture On Cotangent Bundles Of Open Manifold

We give an proof on the Weinstein conjecture on the cotangent bundles of open manifolds. Its proof is based on Gromov's nonlinear Fredholm alternative

J-holomorphic Curves, Legendre Submanifolds and Reeb Chords

In this article, we prove that there exists at least one chord which is characteristic of Reeb vector field connecting a given Legendre submanifold in a closed contact manifold with any contact form

J-holomorphic Curves And Periodic Reeb Orbits

We study the $J-$holomorphic curves in the symplectization of the contact manifolds and prove that there exists at least one periodic Reeb orbits in any closed contact manifold with any contact form by using the well-known Gromov's nonlinear Fredholm alternative for $J-$holomorphic curves. As a corollary, we give a complete solution on the well-known Weinstein conjecture.Comment: arXiv admin note: substantial text overlap with arXiv:math/0004038, arXiv:math/0309258, arXiv:math/0309270, arXiv:math/0309205, arXiv:math/000403

A Proof On Arnold Chord Conjecture In Cotangent Bundles

We prove the Arnold chord conjecture on cotangent bundles of open manifold by Gromov's nonlinear Fredholm alternative for $J-$holomorphic curves

Proofs On Arnold Conjectures

In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture

Pre-Lagrangian Submanifolds in Contact Manifolds

In this paper we prove that there does not exists any closed Pre-Lagrangian submanifolds in any closed contact manifolds by using the $J-$holomorphic curves and Gromov's nonlinear Fredholm alternative