Rotary Diffeomorphism onto Manifolds with Affine Connection

Abstract

In this paper we will introduce a newly found knowledge above the existence and the uniqueness of isoperimetric extremals of rotation on two-dimensional (pseudo-) Riemannian manifolds and on surfaces on Euclidean space. We will obtain the fundamental equations of rotary diffeomorphisms from (pseudo-) Riemannian manifolds for twice-differentiable metric tensors onto manifolds with affine connections.

Rotary mappings of spaces with affine connection

“…Chudá, Mikeš and Sochor [4] also proved that (pseudo-) Riemannian manifold V 2 admits rotary mapping ontoĀ 2 if and only if in V 2 holds equation…”

“…This definition which was formulated by Leiko [12] was later generalized as follows, see [4]. A diffeomorphism f : V 2 →Ā 2 is called rotary mapping if any geodesic on manifoldĀ 2 with affine connection∇ is mapped onto isoperimetric extremal of rotation on two-dimensional (pseudo-) Riemanninan manifold V 2 .…”

“…Later, some new properties were proved, see [4]: When V 2 admits rotary mapping f ontoĀ 2 then if V 2 andĀ 2 in common coordinate system belong differentiability class C 2 and C 1 , respectively, then Gaussian curvature K on V 2 is differentiable. As a result authors formulated new theorem: Rotary diffeomorphism V 2 →Ā 2 does not exist if Gaussian curvature K C 1 .…”

“…Equations of these extremals of rotation were later specified in work [24]. Another contribution to this topic can be found in [4], where authors refined requirements for spaces which admit rotary mapping.…”

“…for any tangent vector X, where ∇ is the Levi-Civita connection, K is the Gaussian curvature, A is a linear form for which A(X) = (X, θ), is a metric tensor, and ν is a function on V 2 . In [4] Chudá, Mikeš and Sochor stated that for any two-dimmensional (pseudo-) Riemannian space V 2 where exist vector fields satisfying the conditions (1) it is possible to construct the space with affine connection A 2 which admits rotary mapping onto V 2 .…”