PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 35, No. 1. September 1972 CLOSED CONVEX HYPERSURFACES WITH SECOND FUNDAMENTAL FORM OF CONSTANT CURVATURE ROLF SCHNEIDER Abstract. It is shown that the Euclidean spheres are the only closed hypersurfaces in Euclidean space on which the second fundamental form defines a (nondegenerate) Riemannian metric of constant curvature. An oriented hypersurface (always assumed to be sufficiently smooth) in (/i+l)-dimensional Euclidean space (n^2) will be called closed if it is compact and without boundary, and convex if its second fundamental form II is positive definite. On a convex hypersurface II defines a Riemannian metric, and it is a natural question to ask for the relations between curvature properties of this metric and the geometrical shape of the hypersurface. We shall prove: Theorem 1. A closed convex hypersurface on which the second fundamental form is of constant Riemannian curvature has to be a Euclidean sphere. For a convex hypersurface S under consideration, let Rn denote the scalar curvature of II and let An be the total area of ¿7corresponding to the metric II. \f Sis closed, it is known to be diffeomorphic to a sphere, and if furthermore II is a metric of constant curvature, the Riemannian space (S, II) must be globally isometric to a Euclidean sphere of radius r, say. In this case we have Ru=n(n— \)r~2 and Au = wnrn, where o;„ denotes the total area of the «-dimensional unit sphere, hence the equality Ru = n(n— \)(conA1i)itn holds. Thus Theorem 1 is a consequence of the following stronger theorem, in which II need not be assumed to be of constant curvature. Theorem 2. A closed convex hypersurface in Euclidean space which satisfies (1) Ru <: n(n - l)(ojnATÎfn has to be a Euclidean sphere. Received by the editors October 29, 1971. AMS 1970 subject classifications. Primary 53A05, 53C40; Secondary 53C20. Key words and phrases. Second fundamental form, closed convex hypersurface, Riemannian metric of constant curvature, scalar curvature, Euclidean sphere. (ç, American 230 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Mathematical Society 1972 231 closed convex hypersurfaces Of course, for «=2 this assertion is not really stronger than that of Theorem 1, since the above inequality and the Gauss-Bonnet theorem (applied to the Riemannian space {S, II)) lead at once to Ä[[=const. Proof of Theorem 2. Let K and H denote Gauss-Kronecker curvature and mean curvature of S, respectively. Let V[l be the first Beltrami differential operator (square of the gradient) with respect to II. Then the identity (2) Ä„ = n(n - \)H + P - {2K)~2V„K holds, where P is some nonnegative function (to be explained later). Deferring the proof of (2) until later, we deduce the inequality Ru ^ n{n - l)K1/n - {2K)~2VUK as a consequence of F^O and H"^.K. Now on the closed hypersurface S the function K attains a maximum. In a point where this happens we have VIIA'=0 and hence Ru^.n{n—\)K1/n. Together with the assumption (1) this leads to (3) K1/2 ^ conATl Since this inequality holds at a point where K attains its maximum, it holds generally on S. Now let dAIt dAu denote the area elements on S with respect to the first fundamental form I and the second fundamental form II, respectively. Then we have K112dA¡=dAu, since K equals the quotient of the determinants of II and I. Integrating over all of S, we deduce, from (3), [k dAx ^ [conAl-¡Kll2dAl = conAj{ ¡dAu = con. Here the left hand integral is also equal to co„, since it represents the total area of the spherical image of the closed convex surface S. We deduce that the equality sign must hold throughout in (3), hence that the function K is constant. But it is well known that the only closed hypersurfaces with constant K are the Euclidean spheres. It remains to prove equation (2). Here P has the following meaning: Let Tfj denote the difference tensor of the Levi-Civita connections with respect to I and II. In the following we shall use the tensor bit of the second fundamental form as the fundamental tensor for "raising and lowering the indices" in the sense of classical tensor notation. Then P is defined by P—TUkT'ik, from which its nonnegativity is obvious. If VI[ denotes covariant differentiation with respect to II, a direct computation gives (Eisenhart [2, p. 33]) Vi\-fk I ' ij ~ rj\l-pk _ ñk V ; » ¡I — "-ijl nk ~~ Kijl i T -rm^rk ' H J mi ~ License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use rr m-r k l lil mli 232 ROLF SCHNEIDER [September where Rkn, R*n are the curvature tensors of II and I, respectively. Contracting once and transvecting with ba, we arrive at (4) Rn = bilRu+ Vffl - VJ'77- T''T4; + TZTZ where Rn is the Ricci tensor of I. Let V1 denote covariant differentiation with respect to I Expressing Vj6H and V"è„ (=0) by means of Christoffel symbols we get Ttkj+ Tjki= —Vlbtj. By the Codazzi equations the right-hand side is symmetric in all three indices, hence Tm (which is symmetric in the first two indices) is symmetric. Therefore equality (4) simplifies to Rn = bilRil+P~(2K)-2VuK, where the formula 2Ti„j = V,„log det I - Vmlog det II = -Vm log K has been used. Finally, the identity bllRn=n(n—l)H the Gauss equations Rkm — gkm(bmj°u is a consequence of — bmlbi¡), where gkl" is the inverse tensor of I. Remark 1. The identity (2) could also be deduced from similar computations by Gardner [4, formula (21)], or from a more general equation in [5, formula (2.23)]; for n = 2 one should also compare E. Cartan [1, p. 11] and Erard [3. p. 7]. For the reader's convenience we gave the short proof in full. Remark 2. If on a (not necessarily closed) convex hypersurface we have R¡¡=1 and A'=l identically, then equation (2) leads to P=0 and hence to V^,;=0. It follows (Simon-Weinstein [6, Corollary 2.1 ], that the hypersurface lies on a sphere. This remark generalizes a result of Erard [3, p. 31], who treated the two-dimensional case. Remark 3. Theorem 1 is, at least for n—2, really a global theorem: There exist convex surfaces in three-dimensional space which are IIisometric to pieces of spheres, but are themselves not pieces of spheres. Such examples have been discovered by Erard [3]. Remark 4. K. Voss [8] has proved that the sphere in three-dimensional space is infinitesimally 11-rigid, i.e. every infinitesimal deformation of the 2-sphere which leaves its second fundamental form stationary has to be an infinitesimal motion. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 233 closed convex hypersurfaces 1972] Remark 5. It would be interesting to know whether there exists, for the second fundamental form, a rigidity theorem analogous to CohnVossen's theorem on isometric ovaloids: Are two closed convex hypersurfaces congruent if their second fundamental forms coincide under some diffeomorphism ? Theorem 1 gives an affirmative answer for the case where one of the hypersurfaces is a sphere. For «=2 closed convex surfaces are known to be congruent if not only the second fundamental forms are the same, but also certain functions of the principal curvatures coincide (e.g., their product—which is a well-known theorem of Grove), see Waiden [9]. Infinitesimal analogues have been proved by Simon [7]. Remark 6. The form K~1/{n+2)Uhas the remarkable property of being invariant under unimodular affine transformations, therefore it serves as one of the fundamental forms in affine differential geometry. It has been shown that ellipsoids are the only closed convex hypersurfaces on which this form is of constant curvature [5, Satz 4.6]. References 1. E. Cartan, Les surfaces qui admettent une seconde forme fondamentale donnée, Bull. Sei. Math (2) 67 (1943), 8-32. MR 7, 30. 2. L. P. Eisenhart, Riemannian geometry, 4th ed., Princeton Univ. Press, Princeton, N.J., 1960. 3. P. J. Erard, Über die zweite Fundamentalform von Flächen im Raum, Doctoral Thesis, ETH, Zürich, 1968. 4. R. B. Gardner, Subscalar pairs of metrics with applications to rigidity and uniqueness of hypersurfaces with a non-degenerate second fundamental form (to appear). 5. R. Schneider, Zur affinen Differentialgeometrie im Grossen. I, Math. Z. 101 (1967), 375-406. MR 36 #3255. 6. U. Simon and A. Weinstein, Anwendungen der de Rhamschen Zerlegung auf Probleme der lokalen Flächentheorie, Manuscripta Math. 1 (1969), 139-146. MR 39 #7538. 7. U. Simon, II-Verlegungen von Eiflächen,Arch. Math. 22 (1971), 319-324. 8. K. Voss, Isometrie von Flächen bezüglich der zweiten Fundamentalform, Nachr. Österr. Math. Ges. 91 (1970), 73. 9. R. Waiden, Eindeutigkeitssätze für ll-isometrischc Eiflächen, Math. Z. 120 (1971), 143-147. Deparimentof Mathematics, Technische Universität License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Berlin, Berlin, Germany

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