We investigate totally umbilical submanifolds in manifolds satisfying some curvature conditions of either recurrent or pseudosymmetry type in the sense of Ryszard Deszcz and derive the respective condition for submanifolds. We also prove some relations involving the mean curvature and the Weyl conformal curvature tensor of submanifolds. Some examples are discussed.
Geometry of traceless cubic forms is studied. It is shown that the traceless part of the cubic form on a statistical manifold determines a conformal-projective equivalence class of statistical manifolds. This conformal-projective equivalence on statistical manifolds is a natural generalization of conformal equivalence on Riemannian manifolds. As an application, Tchebychev type immersions in centroaffine immersions of codimension two are studied.
In the homogeneous space Sol, a translation surface is parametrized by , where and are curves contained in coordinate planes. In this article, we study translation invariant surfaces in , which has finite type immersion.
We introduce the notions of (extrinsic) locally transversally symmetric immersions and submanifolds in a Riemannian manifold equipped with a unit Killing vector field as analogues of those of (extrinsic) locally symmetric immersions and submanifolds. We treat their geometric properties, derive several characterizations and give a list of examples.
The connected components of the zero set of any conformal vector field v, in a pseudo-Riemannian manifold (M, g) of arbitrary signature, are of two types, which may be called ‘essential’ and ‘nonessential’. The former consist of points at which v is essential, that is, cannot be turned into a Killing field by a local conformal change of the metric. In a component of the latter type, points at which v is nonessential form a relatively-open dense subset that is at the same time a totally umbilical...
An integrated approach to Lie derivatives of spinors, spinor connections and the gravitational field is presented, in the context of a previously proposed, partly original formulation of a theory of Einstein-Cartan-Maxwell-Dirac fields based on “minimal geometric data”: the needed underlying structure is determined, via geometric constructions, from the unique assumption of a complex vector bundle with 2-dimensional fibers, called a -spinor bundle. Any further considered object is assumed to...