Dae Won Yoon

Dong-Soo Kim

Young Ho Kim

Kim, Dong-Soo

Kim, Young Ho

Yoon, Dae Won

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 352 (2014) 1033–1038Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comGeometry/Differential geometryOn standard imbeddings of hyperbolic spaces in the Minkowski spaceDong-Soo Kim a,1, Young Ho Kim b,2, Dae Won Yoon ca Department of Mathematics, Chonnam National University, Gwangju 500-757, South Koreab Department of Mathematics, Kyungpook National University, Daegu 702-701, South Koreac Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, South Koreaa r t i c l e i n f o a b s t r a c tArticle history:Received 16 July 2013Accepted after revision 4 September 2014Available online 11 October 2014Presented by the Editorial BoardWe establish some characterizations of the standard imbeddings of hyperbolic spaces in the (n + 1)-dimensional Minkowski space Ln+1 with intrinsic and extrinsic properties such as the n-dimensional area of the sections cut off by hyperplanes, the (n + 1)-dimensional volume of regions between parallel hyperplanes, and the n-dimensional surface area of regions between parallel hyperplanes. In the same manner, we give an affirmatively partial answer to Question A suggested in [6], which is for the characterization of hyperspheres in the (n + 1)-dimensional Euclidean space En+1.© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous établissons quelques caractérisations des plongements standard d’espaces hyperbo-liques dans l’espace de Minkowski Ln+1 de dimension n + 1, avec des propriétés intrinsèques et extrinsèques comme la surface n-dimensionnelle des sections coupées par des hyperplans, le volume en n + 1 dimensions de régions entre des hyperplans parallèles et la surface n-dimensionnelle de régions entre des hyperplans parallèles. De la même façon, nous donnons une réponse affirmative partielle à la question A suggérée dans [6], qui concerne la caractérisation d’hypersphères dans l’espace Euclidien En+1 de dimension n + 1.© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionWe consider the (n + 1)-dimensional Minkowski space Ln+1 = Rn+11 with metric ds2 = dx21 + · · · + dx2n − dx2n+1 for x = (x1, · · · , xn+1). Let us denote by Hn(r) ⊂ Ln+1 the spacelike hyperquadric defined by 〈p, p〉 = −r2, pn+1 > 0. Then Hn(r)E-mail addresses: dosokim@chonnam.ac.kr (D.-S. Kim), yhkim@knu.ac.kr (Y.H. Kim), dwyoon@gnu.ac.kr (D.W. Yoon).1 Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0022926).2 Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042298) and supported by Kyungpook National University Research Fund, 2012.http://dx.doi.org/10.1016/j.crma.2014.09.0031631-073X/© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1034 D.-S. Kim et al. / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 1033–1038is a Riemannian space form with constant sectional curvature K = − 1r2, which is called the standard imbedding of the hyperbolic space of curvature K = − 1r2, or simply the hyperbolic space.For a fixed point p ∈ Hn(r), N(p) = −p/r is the timelike unit normal to Hn(r) at p pointing to the concave side of Hn(r). If t > 0 is sufficiently small, let us denote by Φt the hyperplane passing through the point p − tN(p), which is parallel to the tangent hyperplane Φ0 of Hn(r) at p.We denote by Ap(t), V p(t) and S p(t) the n-dimensional area of the section in Φt enclosed by Φt ∩ Hn(r), the (n + 1)-dimensional volume of the region bounded by the hyperbolic space and the hyperplane Φt , and the n-dimensional surface area of the region of Hn(r) between the two hyperplanes Φt and Φ0, respectively.Then, for sufficiently small t > 0, we have the following properties of the hyperbolic space Hn(r).Proposition 1. The hyperbolic space Hn(r) satisfies the following conditions:(V ) : the (n + 1)-dimensional volume V p(t) of the region is independent of the point p,(A) : the n-dimensional area Ap(t) of the section is independent of the point p,(S) : the n-dimensional surface area S p(t) of the region is independent of the point p.Proof. For two points p and q on the hyperbolic space Hn(r), there exists a linear isometry φ : Ln+1 → Ln+1 carrying Hn(r)isometrically to itself, with φ(p) = q ([10, pp. 113–114]).If we denote by Φ ′0 and Φ ′t the tangent hyperplane of Hn(r) at q and the hyperplane passing through q − tN(q), which is parallel to Φ ′0, respectively, then we also have:φ(N(p)) = N(q), φ(Φ0) = Φ ′0, φ(Φt) = Φ ′t .These relations complete the proof. Conversely, it is natural to ask the following question.Question 2. Are there any other convex spacelike hypersurfaces in the (n + 1)-dimensional Minkowski space Ln+1 that satisfy the above properties?We will say that a convex spacelike hypersurface in Ln+1 is strictly convex if the shape operator of the hypersurface is positive definite with respect to the unit normal N pointing to the concave side.In this article, we study strictly convex spacelike hypersurfaces M in the (n + 1)-dimensional Minkowski space Ln+1of which parallel hypersurfaces satisfy the above-mentioned properties. For a constant s, the parallel hypersurface Ms is defined by the set consisting of the points p − sN(p), p ∈ M . It is a well-defined hypersurface, provided s is small enough.We assume that for each s ∈ (−ε, ε), ε > 0, the parallel hypersurface Ms is strictly convex. Then, for a fixed point p ∈ Ms ⊂ Ln+1, Ap(t), V p(t) and S p(t) are defined as above.In Section 3, as a result, we prove the following theorem.Theorem 3. Let M be a strictly convex spacelike hypersurface in the (n + 1)-dimensional Minkowski space Ln+1. We assume that for each s ∈ (−ε, ε), ε > 0, the parallel hypersurface Ms is strictly convex. Suppose that each Ms , s ∈ (−ε, ε) satisfies one of the following conditions:(A) : the n-dimensional area Ap(t) of the section is independent of the point p ∈ Ms ,(V ) : the (n + 1)-dimensional volume V p(t) of the region is independent of the point p ∈ Ms ,(S) : the n-dimensional surface area S p(t) of the region is independent of the point p ∈ Ms .Then, up to isometries of Ln+1, the hypersurface M is an open part of the hyperbolic space Hn(r).Conversely, for the hyperbolic space M = Hn(r) and sufficiently small s, the parallel hypersurface Ms is nothing but the hyperbolic space Ms = Hn(r + s). Hence, it follows from Proposition 1 that each Ms satisfies the conditions (A), (V ) and (S), respectively.Remark 4. It is well known that the convex and complete hypersurfaces in the Euclidean space with positive constant Gauss–Kronecker curvature with respect to the unit normal pointing to the convex side are the hyperspheres [3]. Using this fact, the first and second authors established some characterizations of Euclidean hyperspheres [5,6]. On the other hand, in the (n + 1)-dimensional Minkowski space Ln+1, there exist a lot of convex and complete spacelike hypersurfaces with positive constant Gauss–Kronecker curvature with respect to the unit normal pointing to the concave side [2,4,7].Finally, if we use the same argument as in the proof of Theorem 3, we are able to give a partial affirmative answer to Question A in [6] as follows.D.-S. Kim et al. / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 1033–1038 1035Theorem 5. Let M be a strictly convex hypersurface in the (n + 1)-dimensional Euclidean space En+1. We assume that for each s ∈ (−ε, ε), ε > 0, the parallel hypersurface Ms is strictly convex. Suppose that each Ms , s ∈ (−ε, ε) satisfies one of the following conditions:(A) : the n-dimensional area Ap(t) of the section is independent of the point p ∈ Ms ,(V ) : the (n + 1)-dimensional volume V p(t) of the region is independent of the point p ∈ Ms ,(S) : the n-dimensional surface area S p(t) of the region is independent of the point p ∈ Ms .Then, up to isometries of En+1, the hypersurface M is an open part of the round hypersphere Sn(r).Throughout this article, all objects are smooth and connected, unless otherwise mentioned.2. PreliminariesSuppose that M is a strictly convex spacelike hypersurface in the (n + 1)-dimensional Minkowski space Ln+1 with the timelike unit normal N pointing to the concave side.We may assume that M is locally the graph of a non-negative convex function f : Rn →R. Since M is spacelike, we have |∇ f (x)| < 1, where ∇ f denotes the gradient vector of the function f . Hence the unit normal N to M isN(x, f (x)) = −1W (x)(f1(x), · · · , fn(x),1) = −1W (x)(∇ f (x),1), (2.1)where f i(x) denotes the partial derivative of f with respect to xi , i = 1, · · · , n and W (x) = (1 − |∇ f (x)|2)1/2.The volume density dV on the hypersurface M is given by ([1, p. 3])dV (X1, · · · , Xn) =∣∣det(〈Xi, X j〉)∣∣1/2, (2.2)where Xi = (ei, f i(x)), i = 1, · · · , n is a basis for T p M , p = (x, f (x)) and e1, · · · , en the natural basis of Rn . It is straightfor-ward to show that:det(〈Xi, X j〉) = det⎛⎜⎜⎜⎝1 − f 21 − f1 f2 · · · − f1 fn− f2 f1 1 − f 22 · · · − f2 fn.........− fn f1 − fn f2 · · · 1 − f 2n⎞⎟⎟⎟⎠ = 1 − |∇ f |2 > 0. (2.3)The shape operator L : T p M → T p M with respect to the unit normal N is defined byL(X) = −∇̄X N, (2.4)where ∇̄ is the usual connection on Ln+1. We denote by k1, · · · , kn the eigenvalues of the shape operator L, which are called the principal curvatures of M at p with respect to the unit normal N . The Gauss–Kronecker curvature K is then defined by K = k1 · · ·kn . Hence we get [7]:K = det(L) = det( f i j)W (x)n+2. (2.5)For spacelike surfaces in L3, the intrinsic Gauss curvature is −K .Since M is a strictly convex spacelike hypersurface in the (n + 1)-dimensional Minkowski space Ln+1 with respect to the timelike unit normal N pointing to the concave side, for each i = 1, 2, · · · , n, we have ki > 0.For a fixed point p ∈ M and a sufficiently small t > 0, consider the hyperplane Φt passing through the point p − tN(p), which is parallel to the tangent hyperplane Φ0 of M at p. Let us again denote by Ap(t), V p(t) and S p(t) the n-dimensional area of the section in Φt enclosed by Φt ∩ M , the (n + 1)-dimensional volume of the region bounded by the hypersurface and the hyperplane Φt and the n-dimensional surface area of the region of M between the two hyperplanes Φt and Φ0, respectively.Now, using a Lorentzian motion of Ln+1, we may introduce a coordinate system (x1, x2, · · · , xn+1) of Ln+1 with the origin p, the tangent space of M at p is the hyperplane xn+1 = 0. Hence, M is locally the graph of a non-negative convex function f : Rn → R satisfying f (0) = 0 and ∇ f (0) = 0.Thus, from N(p) = −(0, · · · , 0, 1), for a sufficiently small t > 0 we obtainAp(t) =¨f (x)<t1 dx, (2.6)and1036 D.-S. Kim et al. / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 1033–1038V p(t) =¨f (x)<t{t − f (x)} dx, (2.7)where x = (x1, x2, · · · , xn) and dx = dx1dx2 · · ·dxn . It follows from (2.2) and (2.3) thatS p(t) =¨f (x)<t√1 − ∣∣∇ f (x)∣∣2 dx. (2.8)Note that we also haveV p(t) =¨f (x)<t{t − f (x)} dx=tˆz=0{ ¨f (x)<z1 dx}dz. (2.9)Together with the fundamental theorem of calculus, Eq. (2.6) shows thatV ′p(t) =¨f (x)<t1 dx = Ap(t). (2.10)First of all, we prove (cf. [5,6]) the following lemma.Lemma 6. Suppose that M is a strictly convex spacelike hypersurface in the (n + 1)-dimensional Minkowski space Ln+1. For the unit normal N pointing to the concave side of M , we have the following:1) limt→01(√t)nAp(t) = (√2)nωn√K (p), (2.11)2) limt→01(√t)n+2V p(t) = (√2)n+2ωn(n + 2)√K (p) , (2.12)3) limt→01(√t)nS p(t) = (√2)nωn√K (p), (2.13)where ωn denotes the volume of the n-dimensional unit ball.Proof. Let us denote by x the column vector (x1, x2, · · · , xn)t. Then, we have from Taylor’s formula of f (x) as follows:f (x) = xt Ax + f3(x), (2.14)where A is a symmetric n × n matrix and f3(x) is an O (|x|3) function. Then, the Hessian matrix of f at the origin is given by ( f i j(0)) = 2A. Hence, for the unit normal N to M we have from (2.5)K (p) = det( f i j(0)) = 2n det A. (2.15)By the assumption, every eigenvalue of A is positive and hence, there exists a nonsingular symmetric matrix B satisfyingA = Bt B, (2.16)where Bt denotes the transpose of B . Therefore, we get:f (x) = |Bx|2 + f3(x). (2.17)We consider the decomposition of S p(t) as follows:Ap(t) = S p(t) + T p(t), (2.18)whereAp(t) =¨1 dx (2.6)f (x)<tD.-S. Kim et al. / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 1033–1038 1037andT p(t) =¨f (x)<t(1 −√1 − |∇ f |2) dx. (2.19)Now, the proof of Lemma 8 in [6] shows that (2.11) and (2.12) hold. Hence, it remains to prove (2.13). In order to prove (2.13), it suffices to show thatlimt→01(√t)nT p(t) = 0. (2.20)Note that the following inequality holds0 ≤ T p(t) ≤¨f (x)<t∣∣∇ f (x)∣∣2 dx. (2.21)The function f satisfies∣∣∇ f (x)∣∣2 = 4|Ax|2 + h2(x), (2.22)where h2(x) is an O (|x|2) function. Thus, there exists a positive constant C satisfying in a neighborhood of the origin∣∣∇ f (x)∣∣2 ≤ C |x|2. (2.23)In the same argument as in the proof of Lemma 8 in [6], putting t = ε2 and x = ε y, it follows from (2.21) and (2.23)that0 ≤ 1(√t)nT p(t) ≤ Cε2¨|B y|2+εg3(y)<1|y|2 dy. (2.24)Since the integral of the right-hand side in (2.24) tends toward a constant as ε → 0, by letting t → 0 in (2.24), we get (2.20). This completes the proof. 3. Proofs of Theorems 3 and 5In this section, first of all, we prove Theorem 3.Let M be a strictly convex spacelike hypersurface in the (n + 1)-dimensional Minkowski space Ln+1. We assume that for each s ∈ (−ε, ε), ε > 0, the parallel hypersurface Ms is strictly convex.Suppose that each Ms , s ∈ (−ε, ε) satisfies one of the three conditions (V ), (A) and (S). Then Lemma 6 shows that for each s ∈ (−ε, ε), the Gauss–Kronecker curvature K (s) of the parallel hypersurface Ms is a positive constant.Now, we prove the following lemma (cf. [8]).Lemma 7. Suppose that M is a strictly convex spacelike hypersurface in the (n + 1)-dimensional Minkowski space Ln+1. Then the following statements are equivalent.1) Each parallel hypersurface Ms has constant Gauss–Kronecker curvature.2) The hypersurface M has constant principal curvatures.Proof. At a fixed point p ∈ M , let k1(p), · · · , kn(p) denote the principal curvatures with principal vectors X1, · · · , Xn , which are orthonormal with respect to the metric g of M . Note that the parallel hypersurface Ms is defined by p − sN(p), p ∈ M , which is a hypersurface for sufficiently small s.With respect to the unit normal N , the shape operator Ls of Ms is given by ([8])Ls = (I − sL)−1L, (3.1)where L and I denote the shape operator of M and the identity operator, respectively. Furthermore, we haveLs(Xi(s)) = ki1 − ski Xi(s), (3.2)where {Xi(s)}ni=1 are orthonormal frame with respect to the metric gs of Ms given byXi(s) = ki Xi . (3.3)1 − ski1038 D.-S. Kim et al. / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 1033–1038Suppose that each parallel hypersurface Ms has constant Gauss–Kronecker curvature K (s). It follows from (3.2) thatn∏i=1ki(p)1 − ski(p) = K (s). (3.4)Hence, we haven∑i=1ln ki(p) −n∑i=1ln(1 − ski(p)) = ln K (s). (3.5)By differentiating (3.5) with respect to s, we get:n∑i=1ki(p)1 − ski(p) =K ′(s)K (s), (3.6)which we denote by k(s). Evaluating k(0), k′(0), · · · , k(n−1)(0), we obtain:n∑i=1ki(p) = k(0),n∑i=1ki(p)2 = k′(0), · · · ,n∑i=1ki(p)n = k(n−1)(0). (3.7)These relations show that each principal curvature ki(p) of M is a constant.The converse is obvious. It follows from Lemma 7 that each principal curvature ki(p) of M is constant, that is, M is isoparametric. Hence M has at most two distinct constant principal curvatures ([8,9]). Furthermore, if M has two distinct constant principal curvatures, then one of them is zero, which is a contradiction. Thus, we see that the constant principal curvatures are positive and equal to each other. That is, M is totally umbilic, but not totally geodesic. Therefore, it is an open part of the hyperbolic space Hn(r) ([10, pp. 116–117]).This completes the proof of Theorem 3.Finally, we prove Theorem 5.Let M be a strictly convex hypersurface in the (n + 1)-dimensional Euclidean space En+1 with respect to the unit normal N pointing to the convex side. We assume that for each s ∈ (−ε, ε), ε > 0, the parallel hypersurface Ms defined by p + sN(p)for p ∈ M is strictly convex.Suppose that each Ms , s ∈ (−ε, ε) satisfies one of the conditions (V ), (A) and (S). Then, Lemma 8 in [6] shows that for each s ∈ (−ε, ε), the Gauss–Kronecker curvature K (s) of the parallel hypersurface Ms is a positive constant. Then, we can use Lemma 7 because it holds for hypersurfaces in an Euclidean space. Hence, the hypersurface M is isoparametric. With the same argument as above, we see that M is totally umbilic, but not totally geodesic. This completes the proof of Theorem 5.References[1] H. Anciaux, Minimal Submanifolds in Pseudo-Riemannian Geometry, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, USA, 2011.[2] J.A. Galvez, A. Martinez, F. Milan, Complete constant Gaussian curvature surfaces in the Minkowski space and harmonic diffeomorphisms onto the hyperbolic plane, Tohoku Math. J. 55 (4) (2003) 467–476.[3] P. Hartman, On complete hypersurfaces of nonnegative sectional curvatures and constant mth mean curvature, Trans. Amer. Math. Soc. 245 (1978) 363–374.[4] J.-i. Hano, K. Nomizu, On isometric immersions of the hyperbolic plane into the Lorentz–Minkowski space and the Monge–Ampère equation of a certain type, Math. Ann. 262 (2) (1983) 245–253.[5] D.-S. Kim, Y.H. Kim, Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl. 437 (1) (2012) 113–120.[6] D.-S. Kim, Y.H. Kim, Some characterizations of spheres and elliptic paraboloids II, Linear Algebra Appl. 438 (3) (2013) 1356–1364.[7] A.M. Li, Spacelike hypersurfaces with constant Gauss–Kronecker curvature in the Minkowski space, Arch. Math. (Basel) 64 (6) (1995) 534–551.[8] K. Nomizu, Élie Cartan’s work on isoparametric families of hypersurfaces, in: Differential Geometry, Part 1, Stanford Univ., Stanford, Calif., 1973, in: Proc. Symp. Pure Math., vol. XXVII, Amer. Math. Soc., Providence, RI, USA, 1975, pp. 191–200.[9] K. Nomizu, On isoparametric hypersurfaces in the Lorentzian space forms, Jpn. J. Math. 7 (1) (1981) 217–226.[10] B. O’Neill, Semi-Riemannian Geometry. With Applications to Relativity, Pure Appl. Math., vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.

On standard imbeddings of hyperbolic spaces in the Minkowski space

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On standard imbeddings of hyperbolic spaces in the Minkowski space

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