Search

In this article, the cosymplectic analogue of the symplectic flux homomorphism of a compact connected cosymplectic manifold (M,η,ω)\left(M,\eta ,\omega ) with ∂M=∅\partial M=\varnothing is studied. This is a continuous map with respect to the C0{C}^{0}-metric, whose kernel is connected by smooth arcs and coincides with the subgroup of all weakly Hamiltonian diffeomorphisms. We discuss the cosymplectic analogue of the Weinstein’s chart, and derive that the group Gη,ω(M){G}_{\eta ,\omega }\left(M) of all cosymplectic diffeomorphisms isotopic to the identity map is locally contractible. A study of an analogue of Polterovich’s regularization process for co-Hamiltonian isotopies follows. Finally, we study Moser’s stability theorems for locally conformal cosymplectic manifolds.

We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$ -perturbations of their fundamental forms: For a manifold ${\mathcal M}^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:{\mathcal M}^d\to {\mathcal N}^{d+1}$ , whose pullback metrics and shape operators are arbitrary close in $L^p$ to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold ${\mathcal N}$ , removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.

Abstract The purpose of the present paper is to examine the isometries of almost Ricci–Yamabe solitons. Firstly, the conditions under which a compact gradient almost Ricci–Yamabe soliton is isometric to Euclidean sphere $$S^n(r)$$ S n ( r ) are obtained. Moreover, we have shown that the potential f of a compact gradient almost Ricci–Yamabe soliton agrees with the Hodge–de Rham potential h. Next, we studied complete gradient almost Ricci–Yamabe soliton with $$\alpha \ne 0$$ α ≠ 0 and non-trivial conformal vector field with non-negative scalar curvature and proved that it is either isometric to Euclidean space $$E^n$$ E n or Euclidean sphere $$S^n.$$ S n . Also, solenoidal and torse-forming vector fields are considered. Lastly, some non-trivial examples are constructed to verify the obtained results.

This paper is an introduction to cosymplectic topology. Through it, we study the structures of the group of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosymplectic manifold (M, ω, η) : (i)− we define and present the features of the space of almost cosymplectic vector fields (resp. cosymplectic vector fields); (ii)− we prove by a direct method that the identity component in the group of all cosymplectic diffeomorphisms is C0−closed in the group Diff∞ (M) (a rigidity result), while in the almost cosymplectic case, we prove that the Reeb vector field determines the almost cosymplectic nature of the C0−limit ϕ of a sequence of almost cosymplectic diffeomorphisms (a rigidity result). A sufficient condition based on Reeb’s vector field which guarantees that ϕ is a cosymplectic diffeomorphism is given (a ˛exibility condition), the cosymplectic analogues of the usual symplectic capacity-inequality theorem are derived and the cosymplectic analogue of a result that was proved by Hofer-Zehnder follows.

We present a geometric formula of Poincaré type, which is inspired by a classical work of Sternberg and Zumbrun, and we provide a classification result of stable solutions of linear elliptic problems with nonlinear Robin conditions on Riemannian manifolds with nonnegative Ricci curvature. The result obtained here is a refinement of a result recently established by Bandle, Mastrolia, Monticelli and Punzo.