In the present paper, we investigate Cartan framed generalized null Mannheim curves in the four-dimensional semi-Euclidean space of index two. We construct the Cartan (or Frenet) frames and curvature functions of generalized Mannheim mate curve with the help of curvatures and Cartan frames of generalized null Mannheim curve.

Here we study the quantum evolution in a flat Riemannian manifold. The holomorphic functions are defined on the cotangent bundle of this manifold. We construct Hilbert spaces of holomorphic functions in which the scalar product is defined using the exponential map. The quantum evolution is proposed by means of an infinitesimal propagator and the holomorphic Feynman integral is developed via the exponential map. The integration corresponding to each step of the Feynman integral is performed in the tangent space. Moreover, the method proposed in this paper naturally takes into account paths that must be included in the development of the corresponding Feynman integral. In the last section we apply our quantization method to the case when the configuration space is a space form, and we show the evolution operator as a composition of infinitesimal evolution operators.

Smooth orbifolds can be considered as a natural generalization of smooth manifolds, when model space is not ${\mathbb R}^n,$ but factor space $ {\mathbb R}^n/G$ by a finite group $G$ of diffeomorphisms. In modern theoretical physics, orbifolds are used as string propagation spaces and are motivated by the preference for models on orbifolds over models on ordinary manifolds. Orbifolds arise in the theory of foliations as ``good'' spaces of leaves. It is known that the existence of a proper leaf with a finite holonomy group for a transversally complete Riemannian foliation is a necessary and sufficient condition for the space of leaves to be an orbifold. The purpose of our paper is study the structure of the space of leaves of singular foliation which generated by orbits of Killing vector fields. Geometry of Killing vector fields is a object of many investigations due to their importance in geometry and physics. We will prove that the space of orbits of family of Killing vector fields on Euclidean space is a smooth orbifold.

We consider a special class of symmetrically loaded thin shells of revolution, which in the presence of certain disturbances of the equilibrium deform without bending. The whole family of such surfaces can be regarded as complementary set to the four classes of equilibrium states of non-bending surfaces that we have studied in some depth recently. The new surfaces (with a few exceptions) have not a closed form description in elementary functions. Here we present their explicit parameterizations in terms of elliptic integrals.

Generalized deformed gauge groups are used for investigation of symmetry of general relativity (GR). GR is formulated in generalized reference frames, which are represented by affine frames. The general principle of relativity is extended to the requirement of invariance of the theory with respect to the group $GL^g$ of local linear transformations of affine frames. GR is interpreted as the gauge theory of the gauge group of translations $T^g_M$. The groups $GL^g$ and $T^g_M$ are united into the group $S^g_M$, which is their semidirect product and is the complete symmetry group of GR. By $GL^g$-gauge fixing one can obtain: Einstein gravity, GR in an orthogonal frame or teleparallel equivalent of GR, dilaton gravity, unimodular gravity, etc.