The work is devoted to the study of the asymptotics of the geometric characteristics of smooth manifolds in phase flows. One of the main such characteristics is the Richie curvature of the manifold with respect to the tangent vector

Navrotskyi, H. V.

Orekhov, O. A.

The work is devoted to the study of the asymptotics of the geometric characteristics of smooth manifolds in phase flows.
One of the main such characteristics is the Richie curvature of the manifold with respect to the tangent vector

Electronic Archive of Kyiv Polytechnic Institute

Asymptotics of geometric characteristics of smooth manifolds in the phase flows

UDC 515ASYMPTOTICS OF GEOMETRIC CHARACTERISTICS OF SMOOTHMANIFOLDS IN THE PHASE FLOWSH. V. Navrotskyi1, a, O. A. Orekhov11National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»,Institute of Physics and TechnologyAbstractThe work is devoted to the study of the asymptotics of the geometric characteristics of smooth manifolds in phase flows.One of the main such characteristics is the Richie curvature of the manifold with respect to the tangent vector.Keywords:manifolds, curvature, metricsIntroductionIn this paper we study asymptotic behaviour of thetopological characteristic for riemannian manifolds inthe smooth flows. It is necessary to determine how theRichie curvature changes for the movement along thetrajectory of a smooth ergodic flow on the Riemannianmanifold M .1. Curvature of a Riemannian manifold1.1. Riemannian manifolsContinuous coordinate system on an open sub-set C of the space Rn is a set of functions{x1(y1, ...yn), ...xn(y1, ..yn)}, that define a one-to-oneand mutually continuous mapping ϕ : C → A ⊂ Rn.Smooth tensor field of order (p, q) on a manifold Mis a set of smooth functions TxM → Rp × Rq, set ineach local coordinate system (x) = (x1, ..., xn)Ti1...ipj1...jq= Ti1...ipj1...jq(x) and when moving to a coordinatesystem xα′change by law:Ti′1...i′pj′1...j′q(xα′) =∑i1...ip∑j1...jq∂xi′1∂xi1∂xip∂xi′p∂xj1∂xj′1∂xjq∂xj′qTi1...ipj1...jq(x).Manifolds are the natural higher-dimensional general-ization of curves and surfaces. A Riemannian manifoldis a manifold equipped with a way to measure the lengthof tangent vectors. The manifold M is called Rieman-nian, if at any point x0 ∈M in the tangent space Tx0a positive definite quadratic form is given (metric ten-sor of order (0, 2) g = gij , where gij are the smoothfunctions of x0). When switching from the coordinatesystem xα to the coordinate system yα′ the componentsof the tensor should change as follows:gi′j′ =∑i,j∂xi∂xi′∂xj∂xj′gij .Let M− Riemannian manifold. Let D ⊂M −− bean open subset of M and there is a diffeomorphismanavrotskiy1992@gmail.comϕ : D → U ⊂ Rn. (ϕ,U)− is a map of D.The volume of the area D ⊂M is the number:vol(D) =∫U√g(x),g(x) = det(gij(x)).Let on a smooth manifold M for every smooth atlasin every map there is a set of smooth functions Γkij(Christoffel symbols) which change when transition tonew coordinates as follows:Γi′k′j′ =∂xi′∂xi∂xj∂xj′∂xk∂xk′Γkij +∂xi′∂xi∂2xi∂xj′∂xk′.A phase flow is a measurable one-parameter groupof one-to-one mappings T t, such that T tx : R× S → Sis measurable mapping and for any t, any A ∈ Φμ((T t)−1A)) = μ(A).An example of a phase flow is the rotation group T t ofthe unit circle S2 : dθ(r, θ) = rdt, with the Lebesguemeasure on the circle, (θ, r) are polar coordinates. Ageodesic is a curve in M such that, for any two closeenough points on the curve, the distance between thesetwo points is realized by the curve. For example, theequator and meridians are geodesics of the sphere, butthe parallels are not. Locally, geodesics between twopoints always exist [1].Theorem 1 (Birkhoff-Khinchin ergodic theorem) gτis a smooth flow, which preserves measure μ, (S,Φ, μ)is space with a normalized measure, f ∈ L1(S,Φ, μ).Then for almost all (in the sense of the measure μ)x0 ∈ S:limt→∞1tt∫0f(gτx0)dτ =M(f(x)|I),where I is a σ−algebra of invariants sets of flow g.In case of the ergodic flow:Математичне моделювання та аналiз даних345limt→∞1tt∫0f(gτx0)dτ =∫Sf(x)μ(dx),and the limit does not depend on position x0 [2].1.2. CurvatureLet P,Q− 2 points on the manifold M , connectedby a smooth curve γ(t).γ(0) = P, γ(1) = Q. γ′−field ofvelocities along γ.ξk – components of this field. Differ-entiation operator along the curve γ:γ′ =∑ξk k .Then the curve γ is called geodesic, if:γ′γ′ = 0.Thus follows the Geodesic equation:d2xidt2+ Γiαkdxαdtdxkdt= 0.The solution of such a system is determined by theconditions:xi(0) = P i,dxidt(0) = ai, a ∈ TP .For any v ∈ TP there exists unique geodesic γv suchthat:γv(0) = P,γ′v(0) = v.Then γv(1) = expP (v) is called exponential mapping inpoint P [3].x ∈M . Tx – is a tangent space in point x. {v, w} ⊂Tx.ε, δ > 0. y – is the endpoint of the vector v. d(x, y) =δ. w′− is vector obtained by translation w by v. Thend(expxεw, expyεw′) = lim(δ(1− ε22K(v, w)))as (δ, ε)→ 0. Where K(v, w) is the sectional curvaturein direction (v, w) The Ricci curvature is the value:Ric(v) =1vol(S)∫SK(v, w)dw,where s is the unit sphere in space RN .K – is a compact set. Fs : K → K− a phase flow.Consider the case, when functional Φ is uniformly con-tinuous on K × [0, 1] , and there is the limit:limt→∞1t∫ t0Φ(Fs, 0)ds,then there is the limit:limt→∞1t∫ t0Φ(Fs, e−s)ds.Proof: ∀ε > 0∃δ ∈ R∀r′, r′′ ∈ K × [0, 1] : |r′ − r′′| < δ|Φ(r′)− Φ(r′′)| < εTherefore, if e−s < δ1t∣∣∣∣∫ t0Φ(Fs, e−s)ds−∫ t0Φ(Fs, 0)ds∣∣∣∣ ≤≤ 1t∫ t0∣∣Φ(Fs, e−s)− Φ(Fs, 0)ds∣∣ ds < ε.ConclusionIn this study we used the main results of the theoryof Ricci curvature of riemmanian manifolds to calculatethe change of it in the smooth flows. It is establishedunder what conditions there is a limit for its time aver-ages.References1. А.С. Мищенко, А.Т. Фоменко. Курс дифференци-альной геометрии и топологии. — 2000.2. I.P. Cornfeld, YA.G. Sinai, S.V. Fomin. ErgodicTheory.— 1982.3. Y. Ollivier. A survey of Ricci curvature for metricspaces and Markov chains.— 2020.—P. 31 p.Всеукраїнська науково-практична конференцiя студентiв, аспiрантiв та молодих вчених346

https://ela.kpi.ua/bitstream/123456789/52490/1/%d0%a1_334-336.pdf

Asymptotics of geometric characteristics of smooth manifolds in the phase flows

Abstract

Similar works

Full text

Available Versions

Electronic Archive of Kyiv Polytechnic Institute

Electronic Archive of Kyiv Polytechnic Institute