CLOSED CONVEX HYPERSURFACES WITH SECOND

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