CURVATURE VIA THE DE SITTER S SPACE-TIME

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1 SARAJEVO JOURNAL OF MATHEMATICS Vol.7 (9 (20, 9 0 CURVATURE VIA THE DE SITTER S SPACE-TIME GRACIELA MARÍA DESIDERI Abstract. We define the central curvature and the total central curvature of a closed curve in a Lorentz (n + space. In addition, we obtain estimates for the total central curvatures of spacelike pure polygons. The study is done by using the n-dimensional de Sitter s space-time.. Introduction In the last three decades interest in Lorentzian geometry has increased, [2]. We will concentrate on a curvature of particular interest here: the total central curvature of closed curves. In some papers, the total central curvature is related to Riemannian spaces; but this curvature has not been treated on spaces with indefinite metrics. In [], Thomas F. Banchoff stated that the total central curvature of a closed curve in 3-dimensional Euclidean space refers to the measure of curvedness of a space curve contained in a bounded ball. He obtained this curvature by averaging the total absolute curvature of the image curves under central proection from all points on the sphere, and he showed that the total central curvature agrees with the classical total absolute curvature of the original space. In [5] we defined, by means of integral formulas, the total central curvature in Lorentzian spaces with dimension 2 and 3. The aim of this paper is to generalize the total central curvature for closed curves in the Lorentzian space L n+, n 3. Also, we will show that if f is a polygon then tcc 2 (f = k, where k is the number of its non middle vertices. In order to do that we will first give the readers the definitions of s-spherical and t-spherical image of a polygon in Lorentzian spaces and we will characterize the local support line to a polygon in the Lorentzian plane Mathematics Subect Classification. 53B30, 53C50. Key words and phrases. Central curvature, total central curvature, spherical image, de Sitter s space-time, Lorentzian space.

2 92 GRACIELA MARÍA DESIDERI Last, we will show that if f n is a polygon with m vertices in L n+, n 2, then tcc n+ (f n m. In section two, preliminaries, we will recall the basic notions in Lorentzian geometry. 2. Preliminaries Let x and y be two vectors in the (n+ dimensional vector space R n+. As it is well known, [8], the Lorentzian inner product of x and y is defined by n+ x, y = x y + x i y i. Thus the square ds 2 of an element of arc-length is given by i=2 n+ ds 2 = dx 2 + dx 2 i. The space R n+ equipped with this metric is called a (n + -dimensional Lorentzian space, or Lorentz (n + -space, which has curvature zero. We write L n+ instead of ( R n+, ds. We say that a vector x in L n+ is timelike if x, x < 0, spacelike if x, x > 0, and null if x, x = 0. The null vectors are also said to be lightlike. We consider the time orientation as follows: a timelike vector x is futurepointing if x, e <0 and past-pointing if x, e > 0, where e =(, 0,..., 0. We say that x is orthogonal to y if x, y = 0, x y 0. Let x be a vector in L n+, then x = x, x is called the Lorentzian norm of x. We say that x is a unit vector if x, x = or x, x =. We shall give a hypersurface M in L n+ by expressing its coordinates x i as functions of n parameters in a certain interval. We consider the functions x i to be real functions of real variables. We say that M is a non-lightlike hypersurface if at every p M its tangent space T p M is equipped with a positive definite or Lorentzian metric, [2]. A parametrized curve is called timelike, spacelike or null pure curve if at every point, its tangent vector is timelike, spacelike or null, respectively. Now, we recall the definition of a pure polygon in Lorentzian spaces (cf. [4]. If p,..., p m L n+, a polygon P[p,..., p m ] is called a pure spacelike polygon if p p 2,...., p m p m and p p m are spacelike vectors. Analogous to time orientation, we define a spacelike orientation in Lorentzian plane: a spacelike vector x is a spacelike-for-future vector if x, e 2 > 0, and spacelike-for-past vector if x, e 2 < 0, where e 2 = (0,. i=2

3 CURVATURE VIA THE DE SITTER S SPACE-TIME 93 In L 2, a vertex p i of a pure spacelike polygon is called a middle vertex if p i p i and p i p i+ have the same spacelike orientation, and non middle vertex if p i p i and p i p i+ have { the same spacelike orientation. Classically, we say that S n = x L n+ : x 2 + } n+ i=2 x2 i = is the n dimensional de Sitter s space-time When n =, S is called a Lorentzian circle; in what follows, we will denote: ( S + = { (x, x 2 S : x 2 > 0 }, ( S = { (x, x 2 S : x 2 < 0 }. In [5], we defined the total central curvature of f as follows. Definition. Let (C be a sequence of connected arcs such that : i C ( S,. + ii C C +,. iii 0 < length (C <,. iv lim C = ( S +. If f : S L 2 ( is a continuous map of the circle S in the Lorentzian plane such that f S ( S + =, then the total central curvature of f is defined by tcc 2 (f = lim Υ ξ (f ds C length (C ξ C if this limit exists; in other case, we say that tcc 2 (f =. 3. S-spherical and t-spherical image of a polygon In [7], Milnor related the total curvature of a polygon in R n+ to its spherical polygon. He defined the spherical image of a vector p i+ p i R n+ as q i = p i+ p i p i+ p i Sn, where S n is the unit n dimensional Euclidean sphere. A spherical polygon Q is formed on S n by oining each q i to q i by a great circle of length 0 α i π. If p i, p i+ L n+, the vector p i+ p i is a timelike, spacelike or lightlike vector. Definition 2. Let p i, p i+ L n+. i If p i+ p i is a timelike vector, { then its t-spherical image is q i = p i+ p i p i+ p i Hn 0, where Hn 0 = x L n+ : x 2 + } n+ i=2 x2 i =. ii If p i+ p i is a spacelike vector, then its s-spherical image is q i = p i+ p i p i+ p i Sn. Next, we define the oriented s-spherical image in Lorentzian plane.

4 94 GRACIELA MARÍA DESIDERI Definition 3. Let p i, p i+ L 2 such that p i+ p i is a spacelike vector. i The s for-future spherical image of p i+ p i is { pi+ p i p i+ p i p i p i+ p i+ p i q i + if p i+ p i is spacelike-for-future, = if p i+ p i is spacelike-for-past. ii The s-for-past { spherical image of p i+ p i is pi+ p i qi p = i+ p i if p i+ p i is spacelike-for-past, if p i+ p i is spacelike-for-future. p i p i+ p i+ p i De Sitter s space-time is geodesically complete; however, there are points in the space which cannot be oined to each other by any geodesic. This is in contrast to spaces with a positive definite metric, where geodesic completeness guarantees that any two points of a space can be oined by at least one geodesic, (cf. [6]. Proposition 4. Let P=P[p,..., p m ] be a pure spacelike polygon in Lorentz 3-space. If p,..., p m lie on the plane x = 0, then q i lies on the curve given by x x2 3 =, i. Hence the edge q i q i is spacelike, that is q i oins to q i by a spacelike geodesic curve in S 2. Proof. By definition of s-spherical image. We define the s for-future and s-for-past spherical polygon of a pure spacelike polygon P=P[p,..., p m ] in the Lorentzian plane. Without loss of generality, in what follows we assume that p is a non middle vertex (cf. [4]. Definition 5. Let P=P[p,..., p m ] be a pure spacelike polygon in Lorentzian plane. i The s for-future spherical polygon of P, Q +, is formed on ( S + by oining each q i + to q+ i by a connected arc with finite length. ii The s-for-past spherical polygon of P, Q, is formed on ( S by oining each qi to q i by a connected arc with finite length. In [5], we define a local support line in Lorentzian plane. Definition 6. Let f : S L 2 be a continuous map of the circle S in the Lorentzian plane. A local support line to f at x is a line containing x and bounding a closed half-plane which contains the image of a neighborhood of x in S. We denote the number of local support to f passing through the point ξ by Υ ξ (f. Now we characterize the local support line to a polygon in Lorentzian plane.

5 CURVATURE VIA THE DE SITTER S SPACE-TIME 95 Theorem 7. Let P=P[p,..., p m ] be a pure spacelike polygon in L 2 and let Q=Q[q,..., q tm ] be its s-spherical polygon. Without loss of generality, we choose the orientation such that every q i lie on ( S +. Let ξ ( S +. Then: i The line ξp is a local support line to P if and only if ξ does not lie on the edge q q m. ii If i > and p i is an non middle vertex, then the line ξp i is a local support line to P if and only if ξ does not lie on the edge q i q i. iii If p i is a middle vertex, then the line ξp i is a local support line to P if and only if ξ lies on the edge q i q i. Proof. Let P=P[p,..., p m ] be a pure spacelike polygon in L 2. A line is local line support of P if and only if it contains a vertex p. In the other hand, the sum of non middle angles is equal to sum of middle angles of P, [4]. By definition of middle and non middle angles, and by the definition of q i, i-iii are following. Let us recall that the angle ω i between q i and q i+ is a central angle and it is equal to length of arc oining q i and q i+. 4. Total central curvature of closed curves in L n+ Let us recall that S n Ln+ is not a compact hypersurface and its volume is not finite either. Hence, we first will define the curvature of a continuous map f n with respect to a connected n-region R n S n which has finite volume. In what follows, we consider f n : S L n+ a continuous map of the circle S ( in the Lorentz (n + -space such that x, x > x f n S. We first define the central proection map in L n+. Definition 8. Let T p (S n be the tangent space to Sn at p Sn and let Ln p be the hyperplane parallel to T p (S n through the center of Sn. The central proection map π p : L n+ T p (S n Ln p is given by π p (x = (x x, p p. x, p This proection map is a one to one and onto map, (cf. [3] for more details. Note that π p restricted to S 2 T ( p S 2 gives the stereographic proection in Lorentzian 3-space. Let us remark that L n p and L n are two congruent spaces. In what follows, we consider n 3.

6 96 GRACIELA MARÍA DESIDERI Definition 9. Let R n Sn be a connected region such that 0 < vol(rn <. The central curvature of f n with respect to R n is defined by ( cc n+ fn, R n ( ( = ( cc n πp i n+ f n, R n L n vol R n p dsr n, p R n where R ds n R n = vol(r n =: volume of Rn ; i n+ : { x L n+ : x, x > } L n+ T p (S n is the inclusion map, and π p : L n+ T p (S n Ln p is the central proection map. We now define the total central curvature of f n. Definition 0. Let (R n be a sequence of connected regions such that : i R n Sn,. ii R n Rn +,. iii 0 < vol(r n <,. iv p S n, 0 such that (R n Ln p, 0. v lim R n = Sn. The total central curvature of f n is defined by tcc n+ (f n = lim cc n+ ( fn, R n if this limit exists; in other case, we say that tcc n+ (f n =. The existence of a sequence (R n satisfying the above mentioned conditions is proven by the following Theorem. Theorem. In L n+ there exists a sequence (R n of connected regions with properties i-v of Definition 0. Proof. Let {a } be a sequence of positive real numbers such that: a a >. b a + = a +,. {( } If R n = x,..., x n+ S 2 : a < x < a then (R n is a sequence of connected regions with i-v properties. Lemma 2. Let (R n be a sequence of connected regions as given in Theorem. If (Ri n i is a subsequence of (R n with properties i-v, then lim cc ( n+ fn ; R n = lim cc n+ (f n ; Ri n. i

7 CURVATURE VIA THE DE SITTER S SPACE-TIME 97 Proof. Since the properties i-iv hold for (R n and (R n i i, and (R n i i (R n, then: and i i, 2i such that R n i R n i R n 2i (5. i, i 2 such that R n i R n R n i 2. (5.2 By (5. and (5.2, lim cc n+ (f n ; R n = lim i cc n+(f n ; R n i. In what follows, if R n S n is a non connected region such that R n = i I Ri n is a disoint union and Ri n is a connected region i I, we denote i I cc ( n fn, Ri n ( by ccn fn, R n. Theorem 3. Let (R n and (Rh n h be two sequences of connected regions as given in Theorem, then lim cc n+ (f n ; R n = lim cc n+(f n ; Rh n. h Proof. By properties i-v, h h, 2h such that: a R n h Rh n Rn 2h b (R n h h, (R n 2h h (R n with properties i-v. By a, we have that vol(r n cc n (π p i n+ f n, (R n 2h 2h L n p p R n 2h < < p (Rn2h Rnh cc n (π p i n+ f n, ((R n 2h R n h L2 p + p Rh n p (R n 2h R n h + p R 2 h By b, we have that lim cc n+(f n ; R n lim h p (R n 2h R n h vol(r 2 2h R 2 h ds (R n 2h R n h cc n (π p i n+ f n, (Rh n Ln p + vol(rh n ds R n h cc n (π p i n+ f n, ((R n 2h R n h L n p vol(r n ds h (R n 2h R n h cc n (π p i n+ f n, (R n h Ln n + vol(r n h ds R n h. cc n (π p i n+ f n, ((R n 2h R n h L n p vol(r n ds h (R n 2h R n h + lim h cc n+(f n ; R n h.

8 98 GRACIELA MARÍA DESIDERI On the other hand, cc n (π p i n+ f n, ((R n 2h R n h L n p ds (R n 2h R n h lim h Then, Analogously, p (R n 2h R n h = constant lim h p (R n 2h R n h cc n (π p i n+ f n, ((R n 2h R n h L n p vol(r n ds h (R n 2h R n = 0. h lim cc ( n+ fn ; R n lim cc n (f n ; Rh n. (5.3 h lim cc ( n+ fn ; R n lim cc n+ (f n ; Rh n. (5.4 h By (5.3 and (5.4, we have that lim cc ( n+ fn ; R n = lim cc n+ (f n ; Rh n. h We now study the regions (R n Ln p. Theorem 4. Let p S n and let (Rn be a sequence of connected regions as given in Theorem. If there exist p and (R n h p L n p hp p (R n Ln p p such that lim hp Rh n p = S n and lim vol(rhp+ n Ln p h p vol(rhp n Ln p =, then (Rh n p L n p hp p is a sequence of connected regions in L n p with properties i-v of Theorem. Proof. Let p be a fixed point of S n and let (Rn be a sequence of connected regions as given in Theorem. There exists p such that p R n, p, and there exists (Rh n p L n p hp p (R n Ln p p such that lim hp Rh n p = S n and lim vol(rhp+ n Ln p h p vol(rhp n Ln p =. Hence, h p p, (S hp L 2 p + is a connected region. Also: i Since R n Sn, then (Rn h p L n p (S n Ln p, h p p. ii Since R n Rn +, then (Rn h p L n p (R n h p+ Ln p, h p p. iii Since 0 < vol(r n <, then 0 < vol(rn Ln p <,. In particular, 0 < vol(r n h p L 2 p <, h p p. v Since lim hp R n h p = S n and lim R n hp Sn (Rn h p L n p = (S n Ln p, then lim hp (R n h p L n p = (S n Ln p.

9 CURVATURE VIA THE DE SITTER S SPACE-TIME 99 iv Since that for each point p S n, p such that R n Ln p p and (R n h p L n p hp p (R n Ln p p, then h p p such that (R n h Ln p h h p. By v, we obtain that for each point q (S n Ln p, h q such that (R n h Ln p L n q h h q. We now show a main theorem in L n+. Theorem 5. The total central curvature of f n is given by tcc n+ (f n = lim vol(r n tcc n (π p i n+ f n ds R n. p R n Proof. Let γ p = π p i n+ f n and tcc n+ (f n = lim cc n+ (f n, R n. According to Definitions 9 and 0, we have that: [ lim tccn (γ p cc n (γ p, R n L n p ] ds R n vol(r 2 = lim vol(r n = lim vol(r n p R n p R n p R n [ ] lim cc n(γ p, R n t t p p cc n (γ p, R n L n p ds R n [ ] lim t p cc n (γ p, Rt n p cc n (γ p, R n L n p ds R n. By Theorems 3 and 4, we may assume that (Rt n p tp p = (Rh n p L 2 p hp p. Then, lim [tcc n (γ p cc n (γ p, R n L n p ]ds R n vol(r n = lim vol(r n p R n p R n Hence, by Lemma 2, lim vol(r n [ ] lim cc n (γ p, Rh n h p L n p cc n (γ p, R n L n p ds R n. p p R n [tcc n (γ p cc n (γ p, R n L n p ]ds R n = Total central curvature of spacelike pure polygons Now we obtain the total central curvature for a spacelike pure polygon. Lemma 6. Let P [p,..., p m ] = f ( S be a spacelike pure polygon in L 2. Then, tcc 2 (f is an positive integer number and tcc 2 (f = k

10 00 GRACIELA MARÍA DESIDERI where k denote the number of non middle vertices of P, 2 k m. Proof. Let (C be a sequence of connected arcs according to Definition. If ξ C, the line through ξ and f (x is a local support line to f at x if and only if there exists i such that f (x = p i vertex of P. Let q i = ( x i, xi 2 the s for-future spherical image of p i+ p i, and let q i0 = A = (a, a 2 and q i = B = (b, b 2 such that a = min i m {x i } and b = max i m {x i }. There exists 0 such that the arc AB C, 0. Then we have: ( Υ ξ (f ds C = Υ ξ (f ds AB + Υ ξ (f ds (C AB. ξ C ξ (C AB ξ AB By Theorem 7 we know that Υ ξ (f = k, then tcc 2 (f = lim Υ ξ (f ds C = k. length (C ξ C Corollary 7. If P is a convex pure spacelike polygon, then tcc 2 (f = 2. Proof. Every convex pure spacelike polygon in Lorentzian plane has exactly two non middle vertices, (cf. [4]. Theorem 8. Let P = f n ( S be a pure spacelike polygon with m vertices in L n+, n 2. Then tcc n+ (f n m. Proof. For each p S n, π p (P is a pure spacelike polygon in L n p which has at most m vertices. The inequalities follow from Theorems 4 and 5, and the relationship tcc 3 (f 2 = lim ( tcc 2 (π p i 3 f 2 ds R 2. area R 2 p R 2 References [] T. F. Banchoff, Total central curvature of curves, Duke Math. J., (969, [2] J. K. Been, P. E. Ehrlich and K. L. Easley, Global Lorentzian Geometry, Second Edition, Marcel Dekker, Inc., New York (996. [3] G. S. Birman and G. M. Desideri, Proections of pseudosphere in the Lorentz 3-space, Bull. Korean Math. Soc., 44 (3 (2007, [4] G. M. Desideri, On polygons in Lorentzian plane, C. R. Acad. Bulgare Sci., 60 (0 (2007, [5] G. M. Desideri, Total central curvature of curves in the 3-dimensional Lorentzian space, Actas del VIII Congreso Monteiro (2005,

11 CURVATURE VIA THE DE SITTER S SPACE-TIME 0 [6] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-time, Cambridge University Press, London (976. [7] J. W. Milnor, On the total curvature of knots, Ann. Math., 52 (2 (950, [8] B. O Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York (983. (Received: June 3, 200 (Revised: October 20, 200 Dto. de Matemática Universidad Nacional del Sur Av. Alem 253, 2 o Piso (8000 Bahía Blanca Argentina E mail: graciela.desideri@uns.edu.ar