Newtonian normal shift in multidimensional Riemannian geometry

Abstract

Explicit description for arbitrary Newtonian dynamical system admitting the normal shift in Riemannian manifold of the dimension $n\geq 3$ is found. On the base of this result the kinematics of normal shift of hypersurfaces along trajectories of such system is studied.Comment: AmSTeX, 38 pages, amsppt styl

Newtonian Dynamical Systems Admitting Normal Blow-Up of Points

“…Therefore part of results of [16] are not valid. This error was corrected in Chapter VII of thesis [17] (see also paper [19]). As a result an explicit formula for general solution of complete system of normality equations (3.32) and (5.1) was derived.…”

“…Definition 2.2 introduces new special class of Newtonian dynamical systems. According to the results of [19], it is not empty (see theorem 12.1 in [19]). In present paper we study this new class of dynamical systems introduced by definition 2.2.…”

“…Newtonian dynamical system (1.1) with force field F on Riemannian manifold M is called a system admitting normal blow-up of points if for any point p 0 ∈ M and for arbitrary positive constant ν 0 > 0 initial data (2.13) determine the normal blow-up f t : p 0 → S t of the point p 0 along trajectories of this dynamical system. Definition 2.2 was first formulated in paper [19]. Theorem 2.1 shows that the condition ν(q) = ν 0 = const built into definition 2.1 is absolutely inevitable.…”

“…The idea to consider blow-ups of one-point sets in the framework of normal shift was suggested by A. V. Bolsinov and A. T. Fomenko when author was reporting results of thesis [17] in a seminar at Moscow State University. Partially this idea was realized in [19]. In that paper was shown that Newtonian dynamical systems admitting the normal shift of hypersurfaces are able to implement normal blow-up of any point p 0 in Riemannian manifold M .…”

Global geometric structures associated with dynamical systems admitting normal shift of hypersurfaces in Riemannian manifolds

“…I am grateful to A. S. Mishchenko for the invitation to visit Moscow and for the opportunity to report the results of thesis [19] and succeeding papers [25], [26], and [27] in his seminar at Moscow State University. I am grateful to N. Yu.…”