On the existence of isoperimetric extremals of rotation and the fundamental equations of rotary diffeomorphisms

Abstract

In this paper we study the existence and the uniqueness of isoperimetric extremals of rotation on two-dimensional (pseudo-) Riemannian manifolds and on surfaces on Euclidean space. We find the new form of their equations which is easier than results by S. G. Leiko. He introduced the notion of rotary diffeomorphisms. In this paper we propose a new proof of the fundamental equations of rotary mappings.

Infinitesimal rotary transformation

“…where ψ i is a covector and θ h is a vector. Similarly as in the papers [1,14] we substitute the equations (10) in (9) and we get…”

“…where c is a constant, K(x) is the Gaussian curvature,λ is a tangent vector, and F(x) is an affinor, tensor field of type 1 1 , see [1,14], which satisfies the following conditions…”

“…These equations hold true for any point and any unit vector λ h . Analogically as described in [1,14] from the above mentioned we obtain the equations…”

Rotary mappings of spaces with affine connection

“…The equations of the isoperimetric extremal of rotation were simplified by Mikeš, Stepanova and Sochor [24] to ∇ s λ = c · K · Fλ, where c is a constant, s is the arc length, F is a tensor 1 1 which satisfies the conditions…”

“…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.…”