XV II th International Conference of Geometry, Integrability and Quantization Sveti Konstantin i Elena 2016 On the existence of isoperimetric extremals of rotation and the fundamental equations of rotary diffeomorphisms Hana CHUDÁ, Josef MIKEŠ Tomas Bata University Zlín Palacký University Olomouc 7. června 2016 H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of rotation 7. června 2016 1 / 27
1 The isoperimetric extremals of rotation The isoperimetric extremals of rotation 2 The fundamental equations of rotary diffeomorphisms The fundamental equation of rotary diffeomorphisms H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of rotation 7. června 2016 2 / 27
The isoperimetric extremals of rotation We suppose that V 2 = (M, g) is a (pseudo-) Riemannian manifold belongs to the smoothness class C r if its metric g C r, i.e. its components g ij (x) C r (U) in some local map (U, x) U M. Differentiability class r is equal to 0, 1, 2,...,, ω, where 0, and ω denote continuous, infinitely differentiable, and real analytic functions, respectively. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of rotation 7. června 2016 3 / 27
The isoperimetric extremals of rotation Let l: (s 0, s 1 ) M be a parametric curve with the equation x = x(s), we construct λ = dx/ds the tangent vector and s is the arc length. The following formulas were developed by analogy with the Frenet formulas for a manifold V 2 : s λ = k n and s n = ε ε n k λ, in these equations represents k the Frenet curvature; n is the unit vector field along l, orthogonal to the tangent vector λ, s is an operator of covariant derivative along l with the respect to the Levi-Civita connection ; ε, ε n are constants ±1. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of rotation 7. června 2016 4 / 27
The isoperimetric extremals of rotation That is to say s λ h = dλh ds + λα Γ h αβ (x(s)) dxβ (s), ds s n h = dnh ds + nα Γ h αβ (x(s)) dxβ (s), ds where Γ h αβ are the Christoffel symbols of V 2, i. e. components of Levi-Civita connection ; λ h and n h are components of the vectors λ and n. We suppose that λ and n are non-isotropic vectors and λ, λ = g ij λ i λ j = ε = ±1, n, n = g ij n i n j = ε n = ±1. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of rotation 7. června 2016 5 / 27
The isoperimetric extremals of rotation Remind assertion for the scalar product of the vectors λ, ξ is defined by λ, ξ = g ij λ i ξ j. Denote s[l] = s1 s 0 λ ds and θ[l] = s1 s 0 k(s) ds functionals of length and rotation of the curve l: x = x(s). And λ = g αβ λ α λ β is the length of a vector λ and k(s) is the Frenet curvature. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of rotation 7. června 2016 6 / 27
The isoperimetric extremals of rotation Definition A curve l is called the isoperimetric extremal of rotation if l is extremal of θ[l] and s[l] = const with fixed ends. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of rotation 7. června 2016 7 / 27
The isoperimetric extremals of rotation G.S. Leiko [2] proved Theorem A curve l is an isoperimetric extremal of rotation only and only if, its Frenet curvature k and Gaussian curvature K are proportional where c = const. k = c K, H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of rotation 7. června 2016 8 / 27
The isoperimetric extremals of rotation J. Mikeš, M. Sochor and E. Stepanova proved the following Theorem The equation of isoperimetric extremal of rotation can be written in the form s λ = c K F λ (1) where c = const. It can be easily proved that vector F λ is also a unit vector orthogonal to a unit vector λ and if c = 0 is satisfied then the curve is geodesic. Structure F is the tensor (1, 1) which satisfies the following conditions (in the invariant form) H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of rotation 7. června 2016 9 / 27
The isoperimetric extremals of rotation F 2 = e Id, g(x, F X) = 0, F = 0. for a Riemannian manifold V 2 C 2 is e = 1 and F is a complex structure. for (pseudo-) Riemannian manifold is e = +1 and F is product structure. The tensor F is uniquely defined with using skew-symmetric and covariantly constant discriminant tensor ε. ( Fi h = ε ij g jh 0 1 and ε ij = g 11 g 22 g12 2 1 0 ) H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 10 / 27
The isoperimetric extremals of rotation Analysis of the equation (1) convinces of the validity of the following theorem which generalizes and refines the results by G.S. Leiko. J. Mikeš proved Theorem Let V 2 be a (non-flat) Riemannian manifold of the smoothness class C 3. Then there is precisely one isoperimetric extremal of rotation going through a point x 0 V 2 in a given non-isotropic direction λ 0 T V 2 and constant c. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 11 / 27
Assume to be given two - dimensional (pseudo-) Riemannian manifolds V 2 = (M, g) and V 2 = (M, g) with metrics g and g, Levi-Civita connections and, structures F and F, respectively. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 12 / 27
Definition A diffeomorphism f : V 2 V 2 is called rotary if any geodesic γ is mapped onto isoperimetric extremal of rotation of manifold V 2. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 13 / 27
Assume a rotary diffeomorphism f: V 2 V 2. Since f is a diffeomorphism, we can impose local coordinate system on M and M, respectively, such that locally f: V 2 V 2 maps points onto points with the same coordinates x, and M M. It holds Theorem Let V 2 admits rotary mapping onto V 2. If V 2 and V 2 belong to class C 2, then Gaussian curvature K on manifold V 2 is differentiable. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 14 / 27
Let γ : x = x(s) be a geodesic on V 2 for which the following equation is valid d 2 x h ds 2 + Γ h ij(x(s)) dxi dx j ds ds = 0 (2) and let γ : x = x(s) be an isoperimetric extremal of rotation on V 2 for which the following equation is valid dλ h ds + Γh ij(x(s)) λ i λ j = c K(x(s)) F h i (x(s)) λ i, (3) where Γ h ij and Γ h ij are components of and, parameters s and s are arc lengthes on γ and γ, λ h = dx h (s)/ds and λ h = dx h (s)/d s. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 15 / 27
Evidently s = s(s). We modify equation (2) dλ h ds + Γh ij(x(s)) λ i λ j = ϱ(s) λ h, (4) where ϱ(s) is a certain function of parameter s. We denote P h ij(x) = Γ h ij(x) Γ h ij(x) the deformation tensor of connections and defined by the rotary diffeomorphism. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 16 / 27
By subtraction equations (3) and (4) we obtain P h ij(x)λ i λ j = c K(x(s)) F h i (x(s)) λ i ϱ(s) λ h (5) Because V 2 C 2 from formulas (4) follows that ϱ(s) C 1. After differentiation formulas (5) we obtain that K(x(s)) C 1. And because these properties apply in any direction, then K is differentiable. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 17 / 27
We proved the following Theorem If Gaussian curvature K / C 1 then rotary diffeomorphism f : V 2 V 2 does not exist. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 18 / 27
Assume that for the rotary diffeomorphism f : V 2 V 2 formulas (5) hold. Contracting equations (5) with g hi λ i we obtain e ε c K = F h i λ i P h αβ λα λ β. (6) After subsequent differentiation of equations (6) along the curve, we obtain (7) e ε c K,δ λ δ = ε γh P h αβ,δ λα λ β λ γ λ δ +c K ε γh P h αβ (2F β δ λα λ δ λ γ +F γ δ λα λ β λ δ ). H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 19 / 27
The part of equation (7), which contains the components of the sixth degree, has the form I = I 1 I 2, (8) where I 1 = c K; I 2 = ε γh P h αβ (2F β δ λα λ δ λ γ + F γ δ λα λ β λ δ ). At the( point x) 0 we can metric tensor diagonalize, such that 1 0 g ij =, where ε = ±1 (by the metric signature). 0 ε Thus from condition λ = 1 follows that (λ 2 ) 2 = εe e(λ 1 ) 2. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 20 / 27
With detailed analysis the highest degrees of (λ 1 ) 6 in the equation (8), we get A = 3e C B = 3e D, (9) where A = 3e( 2P 2 12 P 1 11 + ep 2 22), B = 3e(eP 2 11 P 2 22 2P 1 12), C = ( 2P 2 12 P 1 11 + ep 2 22), D = (ep 2 11 P 2 22 2P 1 12). H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 21 / 27
From this it follows A 2 + B 2 = 0, and evidently A = B = 0, and hence we have in this coordinate system 2P 2 12 P 1 11 + ep 2 22 = 0 and ep 2 11 P 2 22 2P 1 12 = 0. We denote ψ 1 = P12 2, ψ 2 = P12 1, θ1 = P22 2 and θ2 = P11 2. We can rewrite the above mentioned formula (5) equivalently to the following tensor equation Pij h = δi h ψ j + δj h ψ i + θ h g ij, (10) where ψ i and θ h are covectors and vector fields. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 22 / 27
On the other hand, we use the same analysis, as in the previous part for this equation θ h = c ε K F h i λ i ε (ϱ 2ψ α λ α ) λ h (11) Further, obtain the following ξ h θ h F h α = e ε c K λ h ε (ϱ 2ψ α λ α ) λ h F h α and after differentiation along curve l and with analysis after the highest degrees we get α θ h λ α θ h θ α λ α θ h K α λ α /K = λ h ( β θ α λ α λ β θ α θ β λ α λ β θ α λ α K β λ β /K and by similar way we obtain the following formulas where ν is a function on V 2. j θ i = θ i (θ j + K j /K) + ν g ij, (12) H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 23 / 27
Theorem The equations P h ij = δ h i ψ j + δ h j ψ i + θ h g ij, (10) j θ i = θ i (θ j + K j /K) + ν g ij, (12) where ψ i, θ h are covector and vector fields, ν is a function on V 2, are necessary and sufficient conditions of rotary diffeomorphism V 2 onto V 2. This proof is straightforward than the one proposed by G.S. Leiko [3]. We note that the above considerations are possible when V 2 C 2 and V 2 C 2. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 24 / 27
The vector field θ i is torse-forming. Under further conditions on differentiability of the metrics it has been proved in [3] that θ i is concircular. Theorem Let f : V 2 V 2 be a rotary diffeomorphism, metrics of (pseudo-) Riemannian manifolds V 2 and V 2 have differentiability class C 2 on own coordinate system; then manifolds V 2 and V 2 are equidistant and isometric of revolution surfaces and metrics are positive definite. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 25 / 27
References A. Gray, Modern differential geometry of curves and surfaces with Mathematica. 2nd ed. Boca Raton, FL: CRC Press, 1997. S. G. Leiko, Conservation laws for spin trajectories generated by isoperimetric extremals of rotation. Gravitation and Theory of Relativity 26 (1988), 117 124. S. G. Leiko, Rotary diffeomorphisms on Euclidean spaces. Mat. Zametki 47:3 (1990), 52 57. J. Mikeš, M. Sochor, E. Stepanova On the existence of isoperimetric extremals of rotation and the fundamental equations of rotary diffeomorphisms. Filomat, (2015). S. M. Minčić, Lj. S. Velimirović, M. S. Stanković., Integrability conditions of derivational equations of a submanifold of a generalized Riemannian space. Applied Mathematics and Computation 226:1 (2014), 3 9. I. V. Potapenko, Leiko network on the surfaces in the Euclidean space E 3. Ukrainian Math. J. 67 (2015), no. 6, 928 937. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 26 / 27
Thank You for Your time. H.Chudá, J. Mikeš (TBU, UP) On the existence of isoperimetric extremals of 7. rotation června 2016 27 / 27