The current paper is devoting to studying Branching-Toeplitz operators on the Hilbert space $\ell^2(A_q)$, where $A_q$ is the rooted homogeneous tree of degree $q\ge 2$. These operators arise from branching-type stationary stochastic processes indexed by rooted tree, and are natural generalizations of Toeplitz operators on classical Hardy spaces. The new findings of this paper are characterizations of basic operator-theoretic properties of branching-Toeplitz operators including boundedness, Brown-Halmos type and Axler-Chang-Sarason-Volberg type results of semi-commutator, spectra, invertibility and Fredholmness.

We show that the twist subgroup $\mathcal{T}_g$ of a nonorientable surface of genus $g$ can be generated by two elements for every odd $g \geq 21$ and even $g \geq 50$. Using these generators, we can also show that $\mathcal{T}_g$ can be generated by two or three commutators depending on $g$ modulo 4. Moreover, we show that $\mathcal{T}_g$ can be generated by three elements if $g \geq 8$. For this general case, the number of commutator generators is either three or four depending on $g$ modulo 4 again.

We would like to define the notion of geodesics for surfaces with singular points. As surfaces with singular points, we consider framed surfaces. Moreover, as curves with singular points, we consider framed curves. Then we give a definition of geodesic as a framed curve of a framed surface. We investigate properties of the geodesic and give examples.

The Weyl curvature is one of the most fundamental quantities in projective Finsler geometry. In this paper, we study a class of Finsler warped product metrics with quadratic Weyl curvature. We give necessary and sufficient conditions of such metrics to be of quadratic Weyl curvature which are non-trivial in the sense that these metrics are not of Weyl type, refining a theorem due to Gabrani-Sevim-Shen. As its application, we construct infinitely many new non-trivial W-quadratic Finsler warped product metrics. In particular, we find non-trivial W-quadratic Finsler metrics which are not Douglas type.

A finite presentation for the subgroup of the mapping class group of a compact non-orientable surface generated by all Dehn twists was given by Stukow [26]. In this paper, we give an infinite presentation for this group, mainly using the presentation given by Stukow [26] and Birman exact sequences on mapping class groups of non-orientable surfaces.

Given a hypercomplex manifold with a rotating vector field (and additional data), we construct a conical hypercomplex manifold. As a consequence, we associate a quaternionic manifold to a hypercomplex manifold of the same dimension with a rotating vector field. This is a generalization of the HK/QK-correspondence. As an application, we show that a quaternionic manifold can be associated to a conical special complex manifold of half its dimension. Furthermore, a projective special complex manifold (with a canonical c-projective structure) associates with a quaternionic manifold. The latter is a generalization of the supergravity c-map. We do also show that the tangent bundle of any special complex manifold carries a canonical Ricci-flat hypercomplex structure, thereby generalizing the rigid c-map.

A pair of mutually orthogonal complementary foliations of a Riemanian manifold $(M, g)$ is considered. A geometry of such the pair is examined for a suitable torsion-free connection arising from the Bott connection. The connection has a number of nice geometric properties, however, it is not a metric one. The metrization leads to a connection with torsion. This metrized connection is a unique affine connection adapted to each of the foliations and with the torsion inducing selfadjoint endomorphisms of the tangent bundles (Theorem 1.1). Investigation of the geometry of the connection (Lemma 5.1) leads to a tensor which is a mixed measure of the “lack of the symmetry” of the Weingarten operator, so it encodes the extrinsic geometry of both foliations. This tensor is also a conformal invariant (Theorem 1.2).