The cobordism ring B∗ of symplectic manifolds defined by V. L. Ginzburg is isomorphic to the Pontrjagin ring of complex-oriented manifolds with free circle actions. This provides an interpretation of the formal group law of complex cobordism as a composition-law on certain asymptotic expansions. An appendix discusses some related questions about toric manifolds.

Introduction. 2 1. Affine Manifolds 9 1.1. Affine Manifolds and Invariants 9 1.2. Affine manifolds with singularities 15 1.3. Polyhedral decompositions 18 1.4. The discrete Legendre transform 33 1.5. Positivity and simplicity 39 2. From polyhedral decompositions to algebraic spaces 48 2.1. The cone picture 49 2.2. The fan picture 52 3. Logarithmic structures 79 3.

We recall Turaev’s theory of shadows of 4-manifolds, and its use to present 3-manifolds. We then prove a calculus for shadows of 3-manifolds which can be viewed as the analogous of Kirby calculus in the shadow world. This calculus has the pleasant feature of being generated only by local moves on the polyhedra.

In this article, the aim is to introduce a para-Sasakian manifold with a canonical paracontact connection. It is shown that φ−conharmonically flat , φ−W2 flat and φ−pseudo projectively flat para-Sasakian manifolds with respect to canonical paracontact connection are all η−Einstein manifolds. Also, we prove that quasi-pseudo projectively flat para-Sasakian manifolds are of constant scalar curvat...

We address the problem of computing the fundamental group of a symplectic S1-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian S1-action. Several examples are presented to illustrate our main results.

We discuss contravariant connections on Poisson manifolds. For vector bundles, the corresponding operational notion of a contravariant derivative had been introduced by I. Vaisman. We show that these connections play an important role in the study of global properties of Poisson manifolds and we use them to define Poisson holonomy and new invariants of Poisson manifolds.

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems of Finsler manifolds under various curvature conditions. As applications, we derive Mckean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize a result on fundamental group due to Milnor to Finsler manifolds.

The main result of this paper is the $G$-homotopy invariance $G$-index signature operator proper co-compact $G$-manifolds. If $G$ manifolds $X$ and $Y$ are equivalent, then we prove that images their operators by map same in $K$-theory $C^{*}$-algebra group $G$. Neither discreteness locally compact nor freeness action on required, so a generalization classical case closed manifolds. Using can d...

Building on previous results [17, 35], we complete the classification of compact oriented Einstein 4-manifolds with $$\det (W^+) > 0$$ . There are, up to diffeomorphism, exactly 15 manifolds that carry such metrics, and, each these manifolds, metrics sweep out one connected component corresponding moduli space.