New metrics found on toric LCS manifolds.
problem Finding compatible complex structures on toric LCS manifolds.
method Proved a bijective correspondence between toric LCS manifolds and pairs (C,a). result Compact toric LCS manifolds have a positive potential.
Reduces LCS manifolds with symplectic actions, preserving conformal structure.
problem Reduction of LCS manifolds with symplectic actions.
method Locally conformally symplectic reduction procedure.
result Compatibility with locally conformally Kähler structure and contact reduction.
The paper characterizes warped product submanifolds in LCS-manifolds.
problem Characterizing warped product submanifolds in LCS-manifolds.
method Analyzing contact CR-warped product and pseudo-slant submanifolds, and constructing examples.
result There do not exist any proper warped product bi-slant submanifolds of LCS-manifolds.
Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.
problem Defining and studying invariants of elliptic curves in locally conformally symplectic manifolds.
method Using J-holomorphic curves and Gromov-Witten theory to define and study invariants. result Found new phenomena in Riemann-Finsler geometry and an analogue of the Weinstein conjecture.
The Weinstein conjecture is extended to a new class of manifolds.
problem Existence of Reeb 2-curves in locally conformally symplectic manifolds.
method Extended Gromov-Witten theory and elliptic curve counts.
result Partial verification of the conjecture in higher dimensions.
Study of Ricci solitons on specific submanifolds.
problem Understanding Ricci solitons on invariant and anti-invariant submanifolds.
method Investigation of Ricci solitons on invariant and anti-invariant submanifolds of (LCS)n-manifolds with respect to both Riemannian and quarter symmetric metric connections. result Characterization of Ricci solitons on these submanifolds.
Durhuus and Jonsson (1995) introduced the class of "locally constructible" (LC) triangulated manifolds and showed that all the LC 2- and 3-manifolds are spheres. We show here that for each d>3 some LC d-manifolds are not spheres. We prove this result by studying how to collapse products of manifolds with exactly one fa…
The paper explores properties of special manifolds related to Yamabe solitons.
problem Characterizing and studying (LCS)n-manifolds with Yamabe solitons. method Analyzing curvature conditions and constructing specific manifolds.
result Characterizations and constructions of (LCS)n-manifolds with Yamabe solitons. Study on totally real submanifolds in (LCS)n-manifolds.
problem Characterizing properties of totally real submanifolds in (LCS)n-manifolds. method Analysis of submanifolds with respect to Levi-Civita and quarter symmetric metric connections.
result Scalar curvature of C-totally real submanifolds is same for both connections.
Study invariant submanifolds in (LCS)n-manifolds with quarter symmetric metric connection.
problem Characterize invariant submanifolds in (LCS)n-manifolds.
method Investigate invariant submanifolds with respect to quarter symmetric metric connection.
result Mean curvature of invariant submanifold is equal to the mean curvature with Levi-Civita connection.
The abstract investigates convexity in locally conformally symplectic geometry.
problem Characterizing and proving convexity in locally conformally symplectic manifolds.
method Geometric characterization and proof of convexity theorems for twisted and symplectic moment maps.
result Established an analog of the symplectic convexity theorem for locally conformally symplectic manifolds.
New methods prove non-squeezing in locally conformal symplectic geometry.
problem Non-squeezing theorem in locally conformal symplectic geometry.
method Generating functions and spectral selectors for lcs Hamiltonian diffeomorphisms.
result Proves a non-squeezing theorem in S1imesR2nimesS1. New t-LC triangulated manifolds are exponentially many.
problem Counting t-LC triangulated manifolds. method Introducing t-LC triangulated manifolds and proving their exponential growth. result There are at most $2^{ rac{d^3}{2}N}$ triangulated 2-LC d-manifolds with N facets. Constructs symplectic structures on product manifolds from LCS structures.
problem Understanding symplectic structures on product manifolds from LCS structures.
method Constructs a R+-torsor of LCS structures on a covering space. result Bounds the number of fixed points of a Hamiltonian isotopy.
We initiate here the study of Gromov-Witten theory of locally conformally symplectic manifolds or $\lcs$ manifolds, $\lcsm$'s for short, which are a natural generalization of both contact and symplectic manifolds. We find that the main new phenomenon (relative to the symplectic case) is the potential existence of holom…