We give a sufficient condition for a curve $\gamma :ℝ\to {ℝ}^n$ to ensure that the $1$-dimensional Hausdorff measure restricted to $\gamma$ is locally monotone.

We prove that the 1-dimensional Hausdorff measure restricted to a simple real analytic curve $\gamma :ℝ\to {ℝ}^N$, $N\ge 2$, is locally 1-monotone.

A differential 1-form on a $(2k+1)$-dimensional manifolds $M$ defines a singular contact structure if the set $S$ of points where the contact condition is not satisfied, $S=\{p\in M:(\omega \wedge {\left(d\omega \right)}^k\left(p\right)=0\}$, is nowhere dense in $M$. Then $S$ is a hypersurface with singularities and the restriction of $\omega$ to $S$ can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation $\left(\omega \right)$ generated by $\omega$ is determined, up to a diffeomorphism, by its restriction to $S$, if we eliminate certain degenerated singularities...

A reflexion space is generalization of a symmetric space introduced by O. Loos in [4]. We generalize locally symmetric spaces to local reflexion spaces in the similar way. We investigate, when local reflexion spaces are equivalently given by a locally flat Cartan connection of certain type.

We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of a 𝕂-analytic curve is a finite-dimensional vector space. We also show that the action of local diffeomorphisms preserving the quasi-homogeneous curve on this vector space is determined by the infinitesimal action of...

We consider the 2D magnetic Prandtl equation in the Prandtl-Hartmann regime in a periodic domain and prove the local existence and uniqueness of solutions by energy methods in a polynomial weighted Sobolev space. On the one hand, we have noted that the $x$-derivative of the pressure $P$ plays a key role in all known results on the existence and uniqueness of solutions to the Prandtl-Hartmann regime equations, in which the case of favorable $P$$...$

We introduce basic characteristic classes and numbers as new invariants for Riemannian foliations. If the ambient Riemannian manifold $M$ is complete, simply connected (or more generally if the foliation is a transversely orientable Killing foliation) and if the space of leaf closures is compact, then the basic characteristic numbers are determined by the infinitesimal dynamical behavior of the foliation at the union of its closed leaves. In fact, they can be computed with an Atiyah-Bott-Berline-Vergne-type...

We show that locally conformal cosymplectic manifolds may be seen as generalized phase spaces of time-dependent Hamiltonian systems. Thus we extend the results of I. Vaisman for the time-dependent case.

A primary Hopf surface is a compact complex surface with universal cover ${ℂ}^2-\left\{\right(0,0\left)\right\}$ and cyclic fundamental group generated by the transformation $(u,v)\mapsto (\alpha u+\lambda v^m,\beta v)$, $m\in \mathbb{Z}$, and $\alpha ,\beta ,\lambda \in ℂ$ such that $\mid \alpha \mid \ge \mid \beta \mid \>1$ and $(\alpha -{\beta }^m)\lambda =0$. Being diffeomorphic with $S^3\times S^1$ Hopf surfaces cannot admit any Kähler metric. However, it was known that for $\lambda =0$ and $\mid \alpha \mid =\mid \beta \mid$ they admit a locally conformally Kähler metric with parallel Lee form. We here provide the construction of a locally conformally Kähler metric with parallel Lee form for all primary Hopf surfaces of class $1$ ($\lambda =0$). We also show...

We use reflections with respect to submanifolds and related geometric results to develop, inspired by the work of Ferus and other authors, in a unified way a local theory of extrinsic symmetric immersions and submanifolds in a general analytic Riemannian manifold and in locally symmetric spaces. In particular we treat the case of real and complex space forms and study additional relations with holomorphic and symplectic reflections when the ambient space is almost Hermitian. The global case is also...