Journal of Geometry and Physics最新文献

Let M be a non-ruled weakly 2-Hopf real hypersurface in a nonflat complex plane whose mean curvature is invariant along the Reeb flow. In this paper, it is proved that M is Levi-flat if and only if M is locally congruent to a strongly 2-Hopf real hypersurface of a special type. As a corollary we present a class of non-ruled Levi-flat real hypersurfaces of dimension three with non-constant mean curvature.

We provide a framework to classify hyperbolic monopoles with continuous symmetries and find a Structure Theorem, greatly simplifying the construction of all those with spherically symmetry. In doing so, we reduce the problem of finding spherically symmetric hyperbolic monopoles to a problem in representation theory. Additionally, we determine constraints on the structure groups of such monopoles. Using these results, we construct novel spherically symmetric $\mathrm{Sp}\left(n\right)$ hyperbolic monopoles.

The paper is devoted to the study of geodesic orbit Riemannian metrics on nilpotent Lie groups. The main result is the construction of continuous families of pairwise non-isomorphic connected and simply connected nilpotent Lie groups, every of which admits geodesic orbit metrics. The minimum dimension of groups in the constructed families is 10.

We study properties of a certain symmetric tensor r induced by the intrinsic torsion of a Riemannian G–structure. This tensor naturally arises in the context of nearly Kähler manifolds and is parallel with respect to the canonical Hermitian connection. In general, we call a G-structure a second order parallel if ${\nabla }^{G}r=0$ for a minimal G–connection ${\nabla }^G$. We show correlation with harmonicity of a G–structure and with G–structures with parallel torsion. An example of second order parallel G–structure is, apart from nearly Kähler manifolds and nearly parallel $G_2$ structures, an α-Kenmotsu manifold.

We study homological mirror symmetry for Hirzebruch surfaces $F_k$ as complex manifolds by using the Strominger-Yau-Zaslow construction of mirror pair and Morse homotopy. For toric Fano surfaces, Futaki-Kajiura and the author proved homological mirror symmetry by using Morse homotopy in [9], [10], [16]. In this paper, we extend Futaki-Kajiura's result of the Hirzebruch surface $F_1$ to $F_k$. We discuss Morse homotopy and show that homological mirror symmetry in the sense above holds true.

Let $(M,J)$ be a compact complex surface. In his paper [9], LeBrun showed that J-invariant solutions of the Einstein-Maxwell equations correspond to conformally Kähler constant scalar curvature metrics whose Ricci tensors are J-invariant. In the present paper, we prove that constant scalar curvature Kähler manifolds of even complex dimension give solutions of Einstein equations with matter fields which we call the Einstein-harmonic equations.

Jet manifolds and vector bundles allow one to employ tools of differential geometry to study differential equations, for example those arising as equations of motions in physics. They are necessary for a geometrical formulation of Lagrangian mechanics and the calculus of variations. It is thus only natural to require their generalization in geometry of $Z$-graded manifolds and vector bundles.

Our aim is to construct the k-th order jet bundle $J_E^k$ of an arbitrary $Z$-graded vector bundle $E$ over an arbitrary $Z$-graded manifold $M$. We do so by directly constructing its sheaf of sections, which allows one to quickly prove all its usual properties. It turns out that it is convenient to start with the construction of the graded vector bundle of k-th order (linear) differential operators $D_E^k$ on $E$. In the process, we discuss (principal) symbol maps and a subclass of differential operators whose symbols correspond to completely symmetric k-vector fields, thus finding a graded version of Atiyah Lie algebroid. Necessary rudiments of geometry of $Z$-graded vector bundles over $Z$-graded manifolds are recalled.

We show that diffeomorphisms of an extended phase space with time, energy, momentum and position degrees of freedom leaving invariant a symplectic 2-form and a degenerate orthogonal metric $d{t}^2$, corresponding to the Newtonian time line element, locally satisfy Hamilton's equations up to the usual canonical transformation on the position-momentum subspace.

We study multisymplectic structures taking values in vector bundles with connections from the viewpoint of the Hamiltonian symmetry. We introduce the notion of bundle-valued n-plectic structures and exhibit some properties of them. In addition, we define bundle-valued homotopy momentum sections for bundle-valued n-plectic manifolds with Lie algebroids to discuss momentum map theories in both cases of quaternionic Kähler manifolds and hyper-Kähler manifolds. Furthermore, we generalize the Marsden-Weinstein-Meyer reduction theorem for symplectic manifolds and construct two kinds of reductions of vector-valued 1-plectic manifolds.