GEODESICS ON THE TANGENT SPHERE BUNDLE OF 3-SPHERE

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1 International Electronic Journal of Geometry Volume 6 No. pp c IEJG GEODESICS ON THE TNGENT SPHERE BUNDLE OF 3-SPHERE ISMET YHN Communicated by Cihan ÖZGÜR bstract. The Sasaki Riemann metric g S on the tangent sphere bundle T S 3 of the unit 3-sphere S 3 is obtained by using the geodesic polar coordinate of S 3. The connection coefficients of the Levi Civita connection of the Sasaki Riemann manifold T S 3, g S are found. Furthermore, a system of differential equations which gives all geodesics of Sasaki Riemann manifold is obtained.. Introduction The unit 3-sphere and its tangent sphere bundle are important issues of the differential geometry which have attracted the interest of physicists as well as mathematicians. The unit 3 sphere has been considered as non-relativistic closed universe model by physicists [7]. ccording to this model, the universe has expanded since Big Bang and this expansion will continue until Big Crunch. In [5], U. Pincall considered Hopf tori in S 3 which is the inverse image of the closed curves on S by helping the Hopf projection p : S 3 S. In [6], Sasaki classified geodesics on the tangent sphere bundles of the unit n- sphere S n and the hyperbolic n-space H n by using Sasaki metric on T S n and T H n. Moreover, he obtained geodesics of horizontal, vertical and oblique types on the tangent sphere bundles of the unit 3-sphere and the unit hyperbolic -space. In [], Klingenberg and Sasaki obtained the Sasaki Riemann metric on T S by using the geodesic polar coordinate of S, and they indicated that the unit vector fields which make a constant angle with the geodesic circles of unit sphere S constitute geodesics of T S. In [] and [3], P. T. Nagy expanded the studies in this field from space forms to Riemann manifolds. He defined a new metric on the tangent sphere bundle of a Date: Received: March 03, 03 and ccepted: June 4, Mathematics Subject Classification. 55R5, 53C5. Key words and phrases. The Tangent Sphere Bundle of 3-Sphere, Geodesics on Tangent Sphere Bundle. This article is the written version of author s plenary talk delivered on September 03-07, 0 at st International Euroasian Conference on Mathematical Sciences and pplications IECMS- 0 at Prishtine, Kosovo. 00

2 GEODESICS ON THE TNGENT SPHERE BUNDLE OF 3-SPHERE 0 Riemann manifold and examined the geometry of the tangent sphere bundle of the Riemann manifold with respect to this new metric. In this paper, the Sasaki Riemann metric g S on the tangent sphere bundle T S 3 of the unit 3 sphere S 3 is obtained by using the geodesic polar coordinates of S 3. The connection coefficients of the Levi Civita connection of the Sasaki Riemann manifold T S 3, g S are calculated. Furthermore, a system of differential equations which gives all geodesics on T S 3, g S is obtained.. The Riemann Manifold S 3, g This section has been developed by using [], [4], and [6]. This section consists of some subjects as the representation with respect to the geodesic polar coordinates of the unit 3 sphere, the induced Riemann metric on S 3, the basis vectors of the tangent vector space at any point of S 3, the Christoffel symbols of S 3, and the differential equations system which gives all geodesics of S 3. Let <, > be positive definite, symmetric, bilinear form in 4 dimensional Euclidean space E 4 defined by. < u, v >= u v + u v + u 3 v 3 + u 4 v 4, for any vectors u, v E 4. S 3 is a surface in E 4 given by. S 3 = { u = x, x, x 3, x 4 :< u, u >=, u E 4}. S 3 is called as the unit 3 sphere in E 4. The unit 3 sphere is given by the following equation.3 x + x + x 3 + x 4 =, with respect to Cartesian coordinate system. The unit 3 sphere is also represented by.4 x = sin ω sin a cos θ, x = sin ω sin a sin θ, x 3 = sin ω cos a, x 4 = cos ω, with respect to the geodesic polar coordinate of S 3 if a curve on S 3 is described by giving the following coordinates as a function of a single parameter t..5 a = at, θ = θt, ω = ωt. In order to find the arc length between infinitely close two points on the unit 3- sphere, the covariant derivations of x, x, x 3, x 4 is used, given by.6 dx = cos ω sin a cos θdω + sin ω cos a cos θda sin ω sin a sin θdθ, dx = cos ω sin a sin θdω + sin ω cos a sin θda + sin ω sin a cos θdθ, dx 3 = cos ω cos adω sin ω sin ada, dx 4 = sin ωdω.

3 0 ISMET YHN The arc length between infinitely close two points on the surface S 3 i.e. x, x, x 3, x 4 and x + dx, x + dx, x 3 + dx 3, x 4 + dx 4 is calculated by.7 ds =< dx, dx, dx 3, dx 4, dx, dx, dx 3, dx 4 > = dx + dx + dx 3 + dx 4. By using the.6, we get for ds the following:.8 ds = dω + sin ω da + sin adθ, and also the matrix representation of the equation in.8 sin ω g ik : 0 sin ω sin a 0, 0 0 where g ik, for i, k {a, θ, ω} is called as induced metric on S 3 from E 4. The inverse matrix of g ik is given by.0 g kj 0 0 sin : ω 0 0. sin a sin ω 0 0 Let e a, θ, ω be any point on the surface S 3 given by. e a, θ, ω = sin ω sin a cos θ, sin ω sin a sin θ, sin ω cos a, cos ω, with respect to standard orthonormal basis of E 4. Since the orthogonal curves on the surface S 3 is described by a = at, θ = θt and ω = ωt, the unit tangent vectors of orthogonal curves passing through the point e a, θ, ω on the surface S 3 can be defined by. f = ω, f 3 = sin ω a, f 4 = sin ω sin a θ. Moreover, the local expressions of the unit tangent vectors f, f 3 and f 4 at the point e a, θ, ω on the surface S 3 are also given by.3 f a, θ, ω = cos ω sin a cos θ, cos ω sin a sin θ, cos ω cos a, sin ω, f 3 a, θ, ω = cos a cos θ, cos a sin θ, cos a, 0, f 4 a, θ, ω = sin θ, cos θ, 0, 0, with respect to standard orthonormal basis of E 4. Thus f, f 3, f 4 are the basis vectors of tangent vector space at any point e a, θ, ω on S 3. Definition.. Let S 3 be the unit 3 sphere in 4-dimensional Euclidean space and let T e S 3 be the tangent vector space consisting of the unit tangent vectors at a point e a, θ, ω on S 3. g is a real valuable function on T e S 3 defined by.4 g : T e S 3 T e S 3 IR X, Y g X, Y = X T g ik Y, where g ik, i, k {a, θ, ω} is the matrix which corresponds to the metric g given by.9. Since g is positive definite, symmetric and bilinear, g must be called as induced Riemann metric on S 3 from E 4.

4 GEODESICS ON THE TNGENT SPHERE BUNDLE OF 3-SPHERE 03 Theorem.. Let S 3 be the unit 3 sphere in 4-dimensional Euclidean space and let {e, f, f 3, f 4 } be another orthonormal basis in Euclidean space E 4. The covariant derivations of e, f, f 3, f 4 are given by de = dωf + sin ωdaf 3 + sin ω sin adθf 4, df = dωe + cos ωdaf 3 + cos ω sin adθf 4, df 3 = sin ωdae cos ωdaf + cos adθf 4, df 4 = sin ω sin adθe cos ω sin adθf cos adθf 3. Proof. We can use the covariant derivations of orthonormal vectors e, f, f 3, f 4 in order to examine the change of the basis vectors on a point in the other infinite closer of each point e a, θ, ω on S 3. The covariant derivatives of these vectors are calculated by using the partial derivation operation as follows: de = e a da + e θ dθ + e ω dω = dωf + sin ωdaf 3 + sin ω sin adθf 4, df = f a da + f θ dθ + f ω dω = dωe + cos ωdaf 3 + cos ω sin adθf 4, df 3 = f 3 a da + f 3 θ dθ + f 3 ω dω = sin ωdae cos ωdaf + cos adθf 4, df 4 = f 4 a da + f 4 θ dθ + f 4 ω dω = sin ω sin adθe cos ω sin adθf cos adθf 3. Theorem.. Let S 3, g be Riemann manifold. Let D be Levi Civita connection of S 3, g and let φ k ij ; i, j, k {a, θ, ω} be Christoffel symbols related to the Riemann metric g. Then the non-zero the Christoffel symbols of S 3, g are given by φ ω aa = sin ω cos ω, φ θ aθ = cot a, φa aω = cot ω, φ a θθ = sin a cos a, φω θθ = sin ω cos ω sin a, φ θ θω = cot ω, where φ k ij = φk ji for all i, j, k {a, θ, ω}. Proof. On the Riemann manifold S 3, g, there is a unique connection D such that D is torsion free and compatible with the Riemann metric g. This connection is called as Levi Civita connection and characterized by the Kozsul formula: g D a θ, ω = a g θ, ω + θ g ω, a ω g a, θ g [ a, θ ], ω + g [ θ, ω ], a + g [ ω, a ], θ, where a = a, θ = θ and ω = ω. Since D is symmetric, [ a, θ ], [ θ, ω ] and [ ω, a ] must be zero. If we get D a θ = φ a aθ a +φ θ aθ θ +φ ω aθ ω, Christoffel symbols are obtained by φ a aθ = gam a g mθ + θ g am m g aθ = 0, φ θ aθ = gθm a g mθ + θ g am m g aθ = cot a, φ ω aθ = gωm a g mθ + θ g am m g aθ = 0, for m {a, θ, ω}. The other Christoffel symbols can be obtained by using the similar method.

5 04 ISMET YHN Theorem.3. Let S 3, g be Riemann manifold and let c : t R ct = at, θt, ωt be a curve on S 3. c is geodesic if and only if the following second order differential equations are provided: a sinh a cosh a θ + cot ωȧ ω = 0, θ + cot aȧ θ + cot ω θ ω = 0, ω sin ω cos ωȧ sin ω cos ω sin a θ = 0. Proof. ct = at, θt, ωt is geodesic if and only if Dċċ is zero. Since ċ is equal to ȧ a + θ θ + ω ω, Dċċ must be equal to: Dȧ a ȧ a + θ θ + ω ω + D θ θ ȧ a + θ θ + ω ω + D ω ω ȧ a + θ θ + ω ω. If we calculate Dċċ in the following way: Dċċ = a sinh a cosh a θ + cot ωȧ ω a + θ + cot aȧ θ + cot ω θ ω θ + ω sin ω cos ωȧ sin ω cos ω sin a θ ω, it can be seen easily that the claim of the theorem is correct. 3. The Sasaki Riemann Manifold T S 3, g S This section consists of some subjects as the expression with the local coordinate function of any point on T S 3, the orthonormal basis at any point on T S 3, the covariant derivations of this orthonormal basis elements, the Sasaki Riemann metric g S on T S 3, the adapted basis and dual basis vectors on T S 3 with respect to g S, the coefficients of the Levi Civita connection of the Sasaki Riemann manifold T S 3, g S, and a system of the differential equations which gives all geodesics of the Sasaki Riemann manifold.. Let T S 3 = e S 3T e S 3 be the disjoint union of the tangent vector spaces including all unit tangent vectors at every point of S 3. Then T S 3 is called as the tangent sphere bundle of S 3. Since S 3 has 3-dimensional manifold structure, T S 3 should be 5 dimensional manifold structure. Let π : T S 3 S 3 be a canonical projection map. ssuming that e is an element of T S 3 at the point e a, θ, ω of S 3. t the same time, e may be considered as a tangent vector in the tangent vector space spanned by the orthonormal frame {f, f 3, f 4 } at the point e a, θ, ω of S 3. If we denote the angle between f 4 and e by δ and the angle between f and the projected vector of e to the tangent plane spanned by the vectors f and f 3 by ϕ, then a, θ, ω, ϕ, δ can be considered as local coordinates for e in π S 3. Therefore, e, e 3 and e 4 have the following local expression: 3. e a, θ, ω, ϕ, δ = cos ϕ sin δf + sin ϕ sin δf 3 + cos δf 4, e 3 a, θ, ω, ϕ, δ = cos ϕ cos δf + sin ϕ cos δf 3 sin δf 4, e 4 a, θ, ω, ϕ, δ = sin ϕf + cos ϕf 3,

6 GEODESICS ON THE TNGENT SPHERE BUNDLE OF 3-SPHERE 05 where e 3 = δ and e 4 = sin δ ϕ are considered as the unit tangent vectors at any point e, e of T S 3 or elements of T S 3. We assume that e, e, e 3, e 4 are the unit orthogonal elements of T S 3. Theorem 3.. Let T S 3 be the tangent sphere bundle of the unit 3 sphere in 4 dimensional Euclidean space and let e, e, e 3, e 4 be the unit orthogonal elements of T S 3. The covariant derivations of these elements are given by where de = w e + w 3 e 3 + w 4 e 4, de = w e + w 3 e 3 + w 4 e 4, de 3 = w 3 e w 3 e + w 34 e 4, de 4 = w 4 e w 4 e w 34 e 3, w = sinh ω sin ϕ sin δda + sin ω sin a cos δdθ + cos ϕ sin δdω, w 3 = sinh ω sin ϕ cos δda + sin ω sin a sin δdθ + cos ϕ cos δdω, w 4 = sinh ω cos ϕda sin ϕdω, w 3 = sin a cos ω cos ϕ cos a sin ϕ dθ + dδ, w 4 = cos ω sin δda + sin a cos ω sin ϕ cos δ cos a cos ϕ cos δ dθ + sin δdϕ, w 34 = cos ω cos δda sin a cos ω sin ϕ sin δ cos a cos ϕ sin δ dθ + cos δdϕ. Proof. We can use the covariant derivations of the unit orthogonal elements e, e, e 3, e 4 in order to examine the change between infinitely close two points on T S 3. The covariant derivatives of these elements can be obtained by using the partial derivation easily. Definition 3.. The -forms providing the equation w ij =< de i, e j > for i, j {,, 3, 4} are called as the connection -forms on the cotangent space T e,e T S 3. Theorem 3.. The square of line element between infinitely close two points on T S 3 is given by 3. dσ = da + dθ + dω + dϕ + dδ + cos ωdadϕ cos a sin ϕ + sin a cos ω cos ϕ dθdδ. Proof. From the study in [] with analogy, we obtained the square of the line element between infinitely close two points on T S 3 as follow: dσ = < de, de > + < de, e 3 > + < de, e 4 > + < de 3, e 4 > = w w + w 3 w 3 + w 4 w 4 + w 3 w 3 + w 4 w 4 + w 34 w 34 = da + dθ + dω + dϕ + dδ + cos ωdadϕ cos a sin ϕ + sin a cos ω cos ϕ dθdδ.

7 06 ISMET YHN The square of the line element between infinitely close two points on T S 3 has the matrix representation as follows: 0 0 cos ω g αβ : for α, β {a, θ, ω, ϕ, δ} cos ω where = cos a sin ϕ + sin a cos ω cos ϕ. The inverse matrix of g αβ is given by csc ω 0 0 cos ω csc ω g βα 0 0 : cos ω csc ω 0 0 csc ω Definition 3.. g S, which has the components g αβ for α, β {a, θ, ω, ϕ, δ}, is called as induced metric on the manifold T S 3. The characteristic vectors of matrix g αβ which has type 5x5 are base vectors of the tangent vector space at point e, e of T S 3 defined by ξ = cos ω a + ϕ, ξ = + cos ω a + ϕ, ξ 3 = ω, ξ 4 = cos ω sin a cos ϕ cos a sin ϕ θ + δ, ξ 5 = + cos ω sin a cos ϕ cos a sin ϕ θ + δ, where k = k for k {a, θ, ω, ϕ, δ}. ξ i; i {,, 3, 4, 5} is called as adapted basis vector of the tangent space T e,e T S 3 with respect to the induced metric g S. If the -form η i ; i {,, 3, 4, 5} provides the following equation: 3.5 η i ξ j = g S ξ i, ξ j = δ i j -form η i is called as adapted dual basis vector of the cotangent space T e,e T S 3 with respect to the induced metric g S.The local expressions of -form η i are given by η = cos ω da + dϕ, η = + cos ω da + dϕ, 3.6 η 3 = dω, η 4 = cos ω sin a cos ϕ cos a sin ϕ dθ + dδ, η 5 = + cos ω sin a cos ϕ cos a sin ϕ dθ + dδ.

8 GEODESICS ON THE TNGENT SPHERE BUNDLE OF 3-SPHERE 07 Theorem 3.3. Let T S 3 be the tangent sphere bundle of the unit 3 sphere and let T e,e T S 3 be a tangent vector space at any point e, e on T S 3. g S, a real valuable function on T e,e T S 3, is a Riemann metric on the manifold T S 3 defined by 3.7 g S : T e,e T S 3 T e,e T S 3 IR X, Y g S X, Y. Proof. Let X = x i ξ i, Y = y j ξ j and Z = z k ξ k for i, j, k {,, 3, 4, 5} be the unit tangent vectors at any point on T S 3 where {ξ, ξ, ξ 3, ξ 4, ξ 5 } is adapted basis of T e,e T S 3. For all X, Y, Z T e,e T S 3 and α, β IR, we get g S αx + βy, Z = g S { α [ ] [ ]} x i ξ i + β y i ξ i, z j ξ j = αx i z i ε i + βy i z i ε i = αg S X, Z + βg S Y, Z. Similarly, we get g S X, α Y + β Z = αg S X, Y + βg S X, Z. Thus, g S is bilinear transformation. Since the following equality is held g S X, Y = g S x i ξ i, y j ξ j = y i x i ε i = g S Y, X. g S must be symmetric map. Finally, g S is a positive definite map because g S provides the following identities: g S X, X = 0 if and only if X = 0 g S X, X > 0 for every X 0. Since g S is positive definite, symmetric and bilinear form, g S must a Riemann metric on the tangent sphere bundle T S 3. Thus, g S is called as the Sasaki Riemann metric. Moreover, T S 3, g S is also called as Sasaki Riemann manifold. Theorem 3.4. Let T S 3, g S be the Sasaki Riemann manifold. Let be Levi Civita connection of T S 3, g S and let Γ γ αβ ; α, β, γ {a, θ, ω, ϕ, δ} be the connection coefficients of the Levi Civita connection i.e. Christoffel symbols related to the matrix g αβ, α, β {a, θ, ω, ϕ, δ}which corresponds to the Sasaki Riemann metric g S. Then the non-zero the Christoffel symbols of T S 3, g S are given by Γ a aω = cot ω, Γa θδ = csc ω a + cos ω ϕ, Γ a ωϕ = csc ω, Γ θ aθ = a, Γ θ aδ = a, Γ θ θω = ω, Γ θ θϕ = ϕ, Γ θ ωδ = ω, Γ θ ϕδ = ϕ, Γ ω aϕ = sin ω, Γω θδ = ω, Γ ϕ aω = csc ω, Γ ϕ θδ = csc ω cos ω a ϕ, Γ ϕ ωϕ = cot ω, Γδ aθ = a, Γ δ aδ = a, Γ δ θω = ω, Γ δ θϕ = ϕ, Γ δ ωδ = ω, Γ δ ϕδ = ϕ.

9 08 ISMET YHN where Γ γ αβ = Γγ βα for all α, β, γ {a, θ, ω, ϕ, δ} and k = k for k {a, θ, ω, ϕ, δ}. Proof. On the Sasaki Riemann manifold T S 3, g S, there is a unique connection such that is torsion free and compatible with Riemann metric g S. This connection is called Levi Civita connection and characterized by the Kozsul formula: g S a θ, ω = a g S θ, ω + θ g S ω, a ω g S a, θ + g S [ a, θ ], ω + g S [ θ, ω ], a + g S [ ω, a ], θ where a = a, θ = θ, ω = ω, ϕ = ϕ and δ = δ. Since Levi Civita connection is symmetric, [ a, θ ], [ θ, ω ], [ ω, a ] must be zero. By using the following identity: a θ = Γ a aθ a + Γ θ aθ θ + Γ ω aθ ω + Γ ϕ aθ ϕ + Γ δ aθ δ, and Kozsul formula, Christoffel symbols are obtained by Γ a aθ = gak a g kθ + θ g ak k g aθ = 0, Γ θ aθ = gθk a g kθ + θ g ak k g aθ = a, Γ ω aθ = gωk a g kθ + θ g ak k g aθ = 0, Γ ϕ aθ = gϕk a g kθ + θ g ak k g aθ = 0, Γ δ aθ = gδk a g kθ + θ g ak k g aθ = a, for k {a, θ, ω, ϕ, δ}. The other Christoffel symbols can be obtained by using the similar method. Theorem 3.5. Let T S 3, g S be the Sasaki Riemann manifold and c : t R ct = at, θt, ωt, ϕt, δt be a curve on the tangent sphere bundle. c is geodesic if and only if the following second order differential equations are provided: a + cot ωȧ ω csc ω a + cos ω ϕ θ δ csc ω ω ϕ = 0, θ { a ȧ θ + a ȧ δ ω θ ω ϕ θ ϕ + ω ω δ + ϕ ϕ δ } = 0, ω sin ωȧ ϕ ω θ δ = 0, ϕ csc ωȧ ω + csc ω cos ω a ϕ θ δ + cot ω ω ϕ = 0, δ + { a ȧ θ a ȧ δ + ω θ ω + ϕ θ ϕ ω ω δ ϕ ϕ δ } = 0. Proof. ct = at, θt, ωt, ϕt, δt is geodesic if and only if ċċ is zero. Since ċ is equal to ȧ a + θ θ + ω ω + ϕ ϕ + δ δ, ċċ is equal to: ȧ a ȧ a + θ θ + ω ω + ϕ ϕ + δ δ + θ θ ȧ a + θ θ + ω ω + ϕ ϕ + δ δ + ω ω ȧ a + θ θ + ω ω + ϕ ϕ + δ δ + ϕ ϕ ȧ a + θ θ + ω ω + ϕ ϕ + ϕ ϕ δ δ + δ δ ȧ a + θ θ + ω ω + ϕ ϕ + δ δ

10 GEODESICS ON THE TNGENT SPHERE BUNDLE OF 3-SPHERE 09 Therefore we get ċċ = a a + ȧ θ a θ + a δ + ȧ ω cot ω a csc ω ϕ +ȧ ϕ sin ω ω + ȧ δ a δ + θ θ + θ ω ω θ + ω δ + θ ϕ ϕ θ + ϕ δ + θ δ { ω ω + csc } ω cos ω a ϕ ϕ + ω ω + ω ϕ csc ω a + cot ω ϕ + ω δ ω θ ω δ + ϕ ϕ + ϕ δ ϕ θ ϕ δ + δ δ If we arrange ċċ in the following way: a + cot ωȧ ω csc ω a + cos ω ϕ θ δ csc ω ω ϕ { + θ a ȧ θ + a ȧ δ ω θ ω ϕ θ ϕ + ω ω δ δ } + ϕ ϕ θ + ω sin ωȧ ϕ ω θ δ ω { + ϕ csc ωȧ ω + csc ω cos ω a ϕ θ δ } + cot ω ω ϕ ϕ { + δ + a ȧ θ a ȧ δ + ω θ ω + ϕ θ ϕ ω ω δ δ } ϕ ϕ δ, it can be seen that the claim of the theorem is true straightforwardly. References [] Kilingenberg, W., and Sasaki, S., On the tangent sphere bundle of a -sphere. Tohuku Math. Journ. 7975, [] Nagy, P. T., On the tangent sphere bundle of a Riemann - manifold. Tohuku Math. Journ. 9977, [3] Nagy, P. T., Geodesics on the tangent sphere bundle of a Riemann manifold, Geometriae Dedicata 7978, [4] O Neill, B., Semi-Riemnn geometry with applications to relativity. cedemic Press, New york, 997. [5] Pinkall, U., Hopf Tori in S 3, Inventiones Mathematicae, 8985, [6] Sasaki, S., Geodesic on the tangent sphere bundles over space forms. J. Reine ngew. Math , [7] Waner S., Levine G. C., Introduction to Differential Geometry and General Relativity, Hofstra University, 005. Pamukkale University, Education Faculty, Department of the Mathematics Education, 0070, Denizli, TURKEY address: iayhan@pau.edu.tr a