ON THE INNER CURVATURE OF THE SECOND FUNDAMENTAL FORM OF A SURFACE IN THE HYPERBOLIC SPACE

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1 ON TE INNER CURVATURE OF TE SECOND FUNDAMENTAL FORM OF A SURFACE IN TE YPERBOLIC SPACE STEVEN VERPOORT The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form It is shown that several conditions on the Gaussian curvature of the second fundamental form can be satisfied only by extrinsic spheres AMS 000 Subject Classification: Primary: 53C4; Secondary: 53C4, 53C45 Key words: second fundamental form, Gaussian curvature, extrinsic sphere 1 INTRODUCTION A substantial number of results in classical differential geometry characterize the spheres among the compact surfaces in the Euclidean three-dimensional space with a positive-definite second fundamental form by means of various curvature conditions The study of the second fundamental form of such surfaces from the metrical point of view has also received considerable attention, and many characterizations of the spheres in terms of the intrinsic curvature of the second fundamental form are known (eg [3] [8] It has been already noticed in [] that several of these results can be generalized to surfaces in the de Sitter space In the present article, some of these theorems will likewise be adapted to surfaces in the three-dimensional hyperbolic space 3 SUMMARY OF TE ADOPTED NOTATION AND USEFUL FORMULAE For a compact surface M 3 endowed with a unit normal vector field N, the corresponding shape operator is given by A : X(M X(M : V V N REV ROUMAINE MAT PURES APPL, 54 (009,,

2 17 Steven Verpoort ere is the Levi-Civita connection of the enveloping space 3, and X(M stands for the collection of all tangent vector fields on the surface M alf of the trace of this operator is defined as the mean curvature of the surface, whereas the determinant of this operator is related to the Gaussian curvature (of its first fundamental form I, which is the restriction of the metric of the enveloping space 3 to the surface through the contracted Gauss equation = K + 1 If the shape operator is a constant multiple of the identity, the surface is said to be an extrinsic sphere The second fundamental form of a compact surface M 3 is defined by II : X(M X(M F(M : (V, W II(V, W = I (A(V, W ere F(M stands for the set of all real-valued functions defined on M On a surface in 3 with strictly positive Gaussian curvature, a unit normal vector field can always be globally chosen in such a way that the second fundamental form becomes positive-definite This will implicitly be assumed for such surfaces The focus of the present article lies on compact (immersed surfaces M in 3 whose second fundamental form is positive-definite, and can thus be seen as a Riemannian metric The Gaussian curvature of this two-dimensional Riemannian space (M, II will be denoted by K II, and is given (see [1] by K II = K K P I (grad K, A grad K 8(K + 1, where P is a certain non-negative function and A stands for the inverse of the shape operator It should also be remarked that Erard s formula (Q (see [4] can be adapted for surfaces in 3 with a positive-definite second fundamental form (Q ( ( I grad, A grad 14 ( I = ( grad ( K II K, grad ( It is straightforward that the above equation is satisfied at the umbilical points of the surface The equation can be established in a region which is free of umbilical points, by expanding both the left- and the right-hand side in terms of the orthonormal basis of principal directions The area element of such a surface will be denoted by dω, whereas the area element of the second fundamental form is given by dω II = dω

3 3 Second fundamental form of a surface in the hyperbolic space CARACTERISATIONS OF EXTRINSIC SPERES In [1] a consideration of the formula for K II resulted in a new proof of the classical result below Theorem 1 A compact surface in 3 has constant Gaussian curvature if and only if it is an extrinsic sphere Some characterizations of extrinsic spheres in 3 by means of the curvature of the second fundamental form can now be given The result below should be compared with the main result of [7] Theorem Let M be a compact surface in the hyperbolic space 3, and assume that the second fundamental form of M is positive-definite The Gaussian curvature of the second fundamental form is constant if and only if M is an extrinsic sphere Proof If a surface in 3 is given, which satisfies the mentioned conditions and most notably has a second fundamental form of constant curvature K II, choose a point p + where K attains its maximal value It is well-known that this maximal value K p+ has to be strictly positive (see, eg, pp 1819 ff of [1] Thus, every q M satisfies K II (q = K II (p+ K K + 1 (p + K K K + 1 (p + (q (where the fact that the function ] 1, + [ R : x x 1+x is increasing has been employed It can be concluded that K II dω II = K II dω K dω = K II dω II This is only possible if the equality K II = K 1+K is satisfied on M In particular, K is constant, and the result follows from the previous theorem Conversely, it is plain that the function K II is constant for an extrinsic sphere The following result is reminiscent to [6] Theorem 3 Assume M 3 is a compact surface in the hyperbolic space with strictly positive Gaussian curvature Then CK = K II (for some C R if and only if M is an extrinsic sphere Proof First of all, it should be remarked that C > 0 This is an immediate consequence of the Gauss Bonnet theorem

4 174 Steven Verpoort 4 Let now p be a point of M where the function = K + 1 attains its minimal value The condition assumed implies that C K (p = K II (p = K K P I (grad K, A grad K 8(K + 1 (p K K + 1 (p Since K (p is strictly positive, we have C (K + 1 (p (p K + 1, (p and the conclusion is that every q M satisfies C K + 1 C K (q (p This means that K II = CK K + 1 K, and by integration there results K II dω II = K II dω K dω Thus, the equality CK II K + 1 = CK = KII is valid, which can only be the case if K is a constant This completes the proof The next lemma will enable us to generalize a result of [8] in the subsequent theorem Lemma 4 Let M 3 be a compact surface with positive-definite second fundamental form If there is a point p where K II has a global minimum while K has a global maximum, then M is an extrinsic sphere Proof It is known that K (p is strictly positive It follows that every point q M satisfies K II (q K II (p K K + 1 (p K K K + 1 (p K + 1 (q A twofold application of the Gauss Bonnet theorem thus gives us that the integral of a non-negative function is zero: 0 = KII K K + 1 dωii Consequently, this integrand vanishes identically, and every inequality in the above reasoning is an equality This means that K II is constant, so that M is an extrinsic sphere

5 5 Second fundamental form of a surface in the hyperbolic space 175 Theorem 5 Let M 3 be a compact surface with a positive-definite second fundamental form Assume that the condition F (K, K II = 0 is fulfilled on M for a function F : R R : (u, v F (u, v such that { Fu > 0, F v 0, Then M is an extrinsic sphere or { Fu 0, F v > 0 (In particular, this condition is satisfied if K = f(k II or K II = f(k for a decreasing function f Proof Suppose that a surface M 3 satisfies the conditions as formulated in the theorem for some function F, including the first set of requirements on F u and F v Let q M be a point where K attains its global maximum, and r M a point where K II attains a global minimum Assume that M is not an extrinsic sphere According to Lemma 4, we necessarily have K (q > K (r, hence 0 = F ( K (q, K II (q > F ( K(r, K II (q F ( K(r, K II (r = 0 This is clearly a contradiction The case in which the second set of conditions is satisfied can be similarly treated The following result is similar to [6] Theorem 6 A compact surface M 3 with strictly positive Gaussian curvature is either an extrinsic sphere, or the function K II K changes sign Proof For a compact surface M 3 with strictly positive Gaussian curvature, which is not an extrinsic sphere, assume that the function K II K does not change sign The latter condition can only be satisfied if this function vanishes identically Let p be a point where the function attains its maximal value Formula (Q reduces at this point p to 0 = ( ( K II K (The symbol = indicates that both sides of an equation should be evaluated at the point p Since M is not an extrinsic sphere, p is non-umbilical, so that it can be concluded that which is clearly a contradiction K II = K > K,

6 176 Steven Verpoort 6 Corollary 7 A compact surface M 3 with strictly positive Gaussian curvature is a sphere if any of the following conditions is satisfied: K (i K II ; K + 1 (ii K II K K + 1 ; (iii K II K The following result generalizes Theorem 9 of [] Theorem 8 Let M 3 be a compact surface with strictly positive Gaussian curvature Assume there are real numbers C, r and s subject to the conditions 0 s 1, r 1, and r + s 1, such that the equation K II = C s K r is satisfied Then M is an extrinsic sphere Proof Let the function ϕ : ] 0, + [ R : x x1 r (1 + x s+1 Remark that ϕ (x 0 for all x ] 0, + [ Furthermore, the constant C has to be strictly positive Let K attains its maximum at a point p + Then one can conclude that (C s K r (p+ = K II p+ = K K P I (grad K, A grad K 8(K + 1 K K + 1 (p + (p + Since s 1 0, we have (K + 1 s 1 s 1 Consequently, C (K + 1 s 1 K r 1 C s 1 K r 1 1, (p + (p + K + 1 (p + hence, for all q M, K1 r C (K + 1 s+1 Thus, we have = ϕ(k (p+ ϕ(k (q = K 1 r (p + (K + 1 s+1 s K K II K + 1 = C s K r K + 1 K (K + 1 s (q

7 7 Second fundamental form of a surface in the hyperbolic space 177 The theorem follows by Corollary 7 The result below is a preparation for Theorem 10, which generalizes a theorem of [5] Lemma 9 Assume a compact surface M 3 has positive-definite second fundamental form If p M is a critical point of, then K II (p K (p Proof Assume for a contradiction that the inequality K II < K holds Since the gradient of vanishes at the point p, formula (Q implies that at the point p we have 0 4( I (grad, grad = ( ( K II K Furthermore, ( is non-negative The above inequality can only be valid if = and, consequently, grad (p vanishes Thus, we get the contradiction K II = K + P K Theorem 10 Let M 3 be a compact surface with strictly positive Gaussian curvature Assume there are real numbers C, r and s subject to the condition 1 r 1, such that the equation K II = C s (K + 1 r K is satisfied Then M is an extrinsic sphere Proof We first show that (1 1 C s+r+1 1 First Case: s + r Let p be a point where attains its minimum, and choose an arbitrary point q M It then follows that (p K K II = C s (K r, (p (p hence 1 C s 1 (K r C s 1++r (p (p C s+r+1 (q Second Case: s + r This follows similarly by investigating a point where attains its maximum

8 178 Steven Verpoort 8 hence also It follows from (1 that (K r 1 r = 1 1+r Cs, K C s (K + 1 r K K + 1 = K II K + 1 The result now follows from Corollary 7 Lemma 11 Let M 3 be a surface with positive-definite second fundamental form If p M is a critical point of, then K II K (p Proof Since p is a critical point of ( grad (p, the formula = grad and similar ones are valid So, at the point p, formula (Q can be rewritten as 1 4 I ( grad, {id A} {A } grad ( = ( K II K Since both operators inside curly brackets in the above formula are positivedefinite, we have 0 ( ( K II K K Assume first that the inequality K II is not satisfied On account of the above inequality, this can only be the case if = But now the rewritten formula (Q reveals that grad = 0, hence also grad = 0 Consequently, K K II = + P K, which contradicts our assumption This completes the proof The next two theorems generalize results of [3] Theorem 1 Let M 3 be a compact surface with strictly positive Gaussian curvature If the equality K II = C s (K + 1 r K is satisfied for real numbers C, s and r, subject to the condition 1 r+s 0, then M is an extrinsic sphere

9 9 Second fundamental form of a surface in the hyperbolic space 179 Proof First, remark that for a critical point p of we have 1 C ( 1 s r Namely, an application of Lemma 11 yields hence C s (K + 1 r K = K II K K + 1, 1 C s 1 (K + 1 r+1 = C s 1 (K + 1 r+1+(s+r (K + 1 (s+r C s 1 (s+r (K + 1 r+1+(s+r ( = C 1 s r (K s+r = C K + 1 It is now an easy consequence that the equality ( 1 s r ( 1 C 1 s r holds on the entire surface M 1 First Case: 1 + s + r 0 Let now p be a point where attains its minimum Then for every point q M we have 1 C 1 s r C 1 s r K + 1 K + 1 (p Second Case: 1 + s + r 0 This follows similarly by investigating a point where attains its maximum It now follows from ( that K C 1 s r (K s+r K = C s (K + 1 r K 1 s r (K s+r K + 1 ( 1 s r = K II K + 1 KII K + 1 The theorem follows from Corollary 7 Lemma 13 Let p be a point of a compact surface M 3 with K > 0 where K II K attains its absolute minimum and attains its absolute maximum Then M is an extrinsic sphere (q Proof Under our assumptions, every point q of M satisfies K II K K II (q K (p (p (q The result follows at once from Corollary 7

10 180 Steven Verpoort 10 Theorem 14 Let M 3 be a compact surface with strictly positive Gaussian curvature Assume that the condition F (, K II K = 0 is fulfilled for a function F : R R : (u, v F (u, v such that { { Fu > 0, Fu 0, or F v 0, F v > 0 Then M is an extrinsic sphere Proof Similar to the proof of Theorem 5 Acknowledgements Work partially supported by the Research Foundation Flanders (Project G04307 REFERENCES [1] JA Aledo, LJ Alías and A Romero, A new proof of Liebmann classical rigidity theorem for surfaces in space forms Rocky Mountain J Math 35 (005, 6, [] JA Aledo and A Romero, Compact spacelike surfaces in 3-dimensional de Sitter space with non-degenerate second fundamental form Differential Geom Appl 19 (003, 1, [3] C Baikoussis and T Koufogiorgos, On convex hypersurfaces in Euclidean space Arch Math (Basel 49 (1987, 4, [4] PJ Erard, Über die zweite Fundamentalform von Flächen im Raum Abhandlung zur Erlangung der Würde eines Doktors der Mathematik der Eidgenössischen Technischen ochschule Zürich Diss No 434, ET Zürich, 1968 [5] T asanis, Characterizations of the sphere by the curvature of the second fundamental form Colloq Math 46 (198, [6] T Koutroufiotis, Two characteristic properties of the sphere Proc Amer Math Soc 44 (1974, [7] R Schneider, Closed convex hypersurfaces with second fundamental form of constant curvature Proc Amer Math Soc 35 (197, [8] U Simon, Characterizations of the sphere by the curvature of the second fundamental form Proc Amer Math Soc 55 (1976, Received 18 May 008 KU Leuven Departement Wiskunde Afdeling Meetkunde Celestijnenlaan 00B bus everlee, Belgium stevenverpoort@wiskuleuvenbe