Annales de l’institut Fourier (AIF)

Dajczer, M.

Rodriguez, L. L.

ANNALES DE L’INSTITUT FOURIER
M. DAJCZER
L. L. RODRIGUEZ
Infinitesimal rigidity of Euclidean submanifolds
Annales de l’institut Fourier, tome 40, no 4 (1990), p. 939-949
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Ann. Inst. Fourier, Grenoble
40, 4 (1990), 939-949.
INFINITESIMAL RIGIDITY
0F EUCLIDEAN SUBMANIFOLDS
by M. DAJCZER and L. RODRIGUEZ
An isometric immersion /: M" -> R^ of a connected n-dimensional
Riemannian manifold into the TV-dimensional Euclidean space is called
rigid if any other isometric immersion g : M"' -> ^N differs from / by
a rigid motion of IR^. A less general rigidity theory deals with the
notion of isometric variations. An isometric variation is a smooth map
F : I x M-> R", 7= [0,1], such that F(0,x) == f{x) and for each t e l ,
Ft(x) = Ff(t,x) is an isometric immersion. The simplest example of an
isometric variation is given by a smooth family of rigid motions of
tR^, i.e., F(t,x) = C(t)f{x) + v(t) where C{t) is an orthogonal trans-
formation of [R^ and v(f) is a vector. Such an isometric variation is
called trivial. If there exists a non-trivial isometric variation we .say that
/ is deformable.
It is easy to see ([GR], p. 53) that the variation vector field
9F f Z7 ^— of F verifiesZ =  
8t ..
(1) <V;.Z,^> = 0
for all Xe TM, where V denotes the standard connection in IR^. If F
is trivial as above, then Z = C'(0)f + i/(0) where C^O) is skew-
symmetric. Conversely, given Z = Cf + v, where C is a skew-symmetric
matrix and u is a constant vector, we have that F(t,x) = e^f(x) + tv
is an isometric variation through rigid motions, i.e., a trivial variation.
Key-words : Euclidean submanifolds - Infinitesimal variations - Infinitesimal rigidity.
A.M.S. Classification : 53B25 - 53C40.
940 M. DAJCZER AND L. RODRIGUEZ
In this paper we deal with a third rigidity notion which is a
linearized version of the second rigidity notion. We say that a smooth
vector fields Z defined along / with values in ^N is an infinitesimal
isometric variation if it verifies (1). An alternative definition of Z is as
the variation vector field of a variation which is isometric only to first
order (see [GR], p. 50). Observe that for any infinitesimal isometric
variation Z, the variation G(t,x) = f(x) + tZ(x) is isometric to first
order to / with — = Z. We say that Z is trivial if it can be writtendt
as Z = Bf + v where B is a skew-symmetric linear transformation in
(R^ and v is a vector. If an immersion only admits trivial infinitesimal
isometric variations we say that it is infinitesimally rigid ; otherwise we
say it is flexible. The following provide generally non-trivial examples
of infinitesimal isometric variations.
Example. — For any normal vector v|/ let A^ denote the second
fundamental form in the direction of \[/. Suppose an immersion
has a normal vector field T| such that at each point A^ = 0. Since
V^T| = — A^X + V^r| = V^r|, where V1 denotes the connection of the
normal bundle, we have that Z = T| satisfies condition (1). In fact, more
generally, we can take Z = Y + T| , where Y is any Killing vector field
on M.
Example. — Let Y be a conformal Killing vector field on a manifold
M\ i.e., <V^r,^> = c(x) <Jr,^>, for every x e M, and /: AT -> R^P
an isometric immersion. Suppose that ^(x) is a normal umbilical direction
which is never totally geodesic, i.e., A^X = 'k{x)X where X- never vanishes.
c(x)Then, if we take Z = Y + —— t., we have that\(x)
<v^z,z> = <v^y,z> + (^x^^x}
\ ^W I
= <v;,y,;o-^<^;w =o,
A^XJ
and Z is an infinitesimal isometric variation. Observe that if M admits
an isometric immersion into S n + p ~ l , then the composition with the
inclusion into Rn+p has always a normal umbilical vector field.
The basic question we consider here is whether the notion of
infinitesimal rigidity differs from the notion of isometric rigidity. Some
INFINITESIMAL RIGIDITY 941
of our results point in the direction that local or global conditions that
imply rigidity also imply infinitesimal rigidity. These types of results
will be treated in Section 1. Another question of rigidity concerns the
existence of infinitesimal isometric variations which satisfy further
conditions. One interesting case is the situation of having infinitesimally
the same Gauss map. In [DG] it is shown that the only possible
examples of isometric variations having the same Gauss map are the
minimal immersions of Kaehler manifolds. We show, in Section 2, that
the only flexible examples are also of this type. We conclude the section
classifying pairs of immersions (not necessarily isometric) which have
the same Gauss map and make a right angle (see [DR2]).
Finally, in Section 3, we use the theory of infinitesimal isometric
variations to obtain a two-parameter family of immersions which are
not rigid.
1. Infinitesimal rigidity.
The following is our basic result.
THEOREM 1. - Let Z be an infinitesimal isometric variation of an
isometric immersion f : M" -^ ffT of a connected manifold M\ Consider
the maps G,: M71 -. OT, t e R, defined as
(2) G,(x)= f(x) + tZ(x).
a) For all te (R, G^ is an immersion and G^ and G-t induce the same
metric.
b) If f is substantial and, for some time to ^ 0, G< and G-< are
congruent in R^ then Z is trivial. ° °
Proof. - Observe that for any tangent vector X,
\\G^X\\2 = ||AT+ ^IIV^ZII2,
since by hypothesis <^,V^Z> = 0. This proves a).
If G^ and G-^ are congruent for some ^ ^ 0, then there exists a
fixed orthogonal transformation T of ^ and a constant vector w € R^
such that
/ + t,Z= T(f-t,Z) + w.
942 M. DAJCZER AND L. RODRIGUEZ
Differentiating with respect to a tangent vector X, we obtain
^+^Z= T(X-t^^Z),
or
(3) t,(T-^I)V^Z= (T-I)X.
If T + I is invertible, then
(4) V^Z = ̂ ,
where ^ = (^+7)~1(T-/), and ^ is skew-symmetric by (4). Since
AT=V;,W), it follows that V^(Z-5/)=0, showing that Z is trivial.
It remains to show that T + I is invertible. Suppose that on the
contrary there exists a vector T| e R^ such that 7^ = - T| . For any
arbitrary tangent vector X, we have using (3),
2<Xr|> = <7XTri> + <^T|> = - <TXT|> + <;r,r|>
= <(7-7);r,r|> = - 4«rV^Z,r|> + <V^Z,T|»
= - ̂ V^T-1^ + <V^Z,T|» = 0.
Since X is arbitrary and T| is constant we have that the immersion is
not substantial, which is a contradiction. This concludes the proof
ofb). D
Recall that Allendoerfer [A] proved that any isometric immersion
into R^ with type number T ^ 3 is rigid. Here we obtain the following
infinitesimal version of this result.
THEOREM 2. — Let f : M" -> (R^ be an isometric immersion mth type
number r(x) ^ 3 for all xeM. Then f is infinitesimally rigid.
Proof. — We first show that any infinitesimal isometric variation is
locally trivial. By theorem la), the immersions Gi and G-< are isometric
and their type number is still greater than or equal to 3 for small t
when restricted to a small open set; consequently, they are locally
congruent and, by Theorem Ib), the variation field Z is locally trivial,
because r(x) ^ 3 implies that the immersion is substantial.
As we saw in the proof of part b) of Theorem 1, if Z is trivial in
two open sets U^ and U^ we have by (2) that Vj^Z = B^X in U^ and
INFINITESIMAL RIGIDITY 943
Vj^Z = ByX in 1/2, where 2?i and ^2 ^r^ skew-symmetric matrices. If
the open sets intersect then (B^—B^)\T M = O ? f°r all x e £/i n £/2. Since
/ is substantial on £/i n U^ the affine tangent spaces TyM, y e U^ r\ U^
generate all of (R^ showing that B^ = B^. We conclude that the variation
is globally trivial. D
Remarks. — 1) The assumption T ^ 3 can be replaced by other
algebraic assumptions which imply rigidity (see [CD]).
2) Goldstein and Ryan [GR] showed that the inclusion 5'" c: R"-^1
is infinitesimally rigid.
3) Any infinitesimally rigid immersion is equi variant.
By a theorem of Sacksteder [Sa], if /, g : M" -^ (R^1, n ^ 3, are
two isometric immersions of a compact manifold as Euclidean
hypersurfaces then / and g are congruent on each connected component
of the set of non-totally geodesic points of /, or on any open connected
totally geodesic set of /. The following is an infinitesimal version of
Sacksteder's theorem. Observe that the condition on the totally geodesic
sets is different from Sacksteder's; we can allow totally geodesic sets
that separate as long as they have empty interior. This last condition
is necessary as Example 1 shows.
THEOREM 3. — Let f : M" ->• Rn+l, n ^ 3, be an isometric immersion
of a compact manifold such that there are no open sets where f is totally
geodesic. Then f is infinitesimally rigid.
Proof. — Suppose that Z is an infinitesimal isometric variation. As
above consider the immersions Gi and G-f We first claim that there
is no open set U where any one of them is totally geodesic. Otherwise,
by Sacksteder's theorem, so would be the other, and consequently /,
since / = l/^G^+G'-^). Fixing a time t + 0, let D denote the set of
totally geodesic points of G^ and let [Ui] denote the family of open
connected subsets of the dense subset M\D. By Sacksteder's theorem
we have that restricted to each £/,, Gt and G-( are congruent.
Consequently, by Theorem 1, we have that there exists a skew-symmetric
linear transformation B^: ̂ n+l -> IR"4'1 such that Z restricted to U^ is
given by
Z=^/+u,,
944 M. DAJCZER AND L. RODRIGUEZ
where v>, is a vector in [R7^1. For each x e M , we can define a unique
linear map C^ in IR"4'1 with the following properties:
1) if X and F are in T^M let <C^, V> = <V^Z, F>,
2) if T| is a normal vector to the immersion / then <C^,T|> =
<V^Z,TI>,
3) C^ is skew-symmetric.
Observe now that on each U,, VjcZ = B,X. Consequently, we have
that at any point x e U,;, Q and 2?, are skew-symmetric matrices which
coincide on f^T^M. Now two skew-symmetric matrices that coincide in
a hyperplane must be equal. Because Z is differentiable, we have that
the map x h-> Cc is also differentiable. However, since this map is locally
constant in the open dense set M\B, it must be globally constant, i.e.,
all the matrices 2?, are equal to a fixed matrix B. Finally, Z = Bf + v,
on each Ui. Therefore, v = Z - Bf is locally constant in M\B, thus
constant in M. D
2. Infinitesimal parallel variations.
We say that Z is an infinitesimal parallel isometric variation if it
satisfies (1) and
(5) <V^Z,T|> =0,
for all Xe TM and all T( e TM1. See Yano [V] for comments on this
notion.
The following theorem classifies completely all immersions which
admit infinitesimal parallel isometric variations (even trivial ones).
THEOREM 4. — Let f: AT -> ^N, n ^ 3, be a locally irreducible
substantial isometric immersion of a connected manifold. Iff admits an
infinitesimal parallel isometric variation Z \vhich is not constant, then M
is Kaehler, and either
i) / is holomorphic and Z = cJf for some complex structure J of ^N
and constant c ; here Z is trivial;
or
ii) / is minimal and Z = cj + v, "where J is the conjugate member of
the associated family of minimal immersions of f, c is a constant and v
is a constant vector.
INFINITESIMAL RIGIDITY 945
Proof. — As in the proof of Theorem 1 consider the isometric
immersions G{ and G-f They have the same Gauss map since
G^X = f^X + t^xZ and by the hypothesis, ^Z is tangent. From [DG]
and [DR1] we have that either G{ and G'_( are congruent or they are
real Kaehlerian associated minimal immersions.
If they are Kaehlerian associated minimal immersions then
G^x) = / ( x ) + r Z ( x ) = ( p
G-f(x) = fW ~ tZ(x) = cos 9(p + sin 9(p,
where q> is the conjugate minimal immersion of (p and 9 is a constant.
Hence
Z = —[(l-cos9)(p-sin9(p]
= h[cos§(p+sin@(p],
for some angle 9 and constant h. Consequently, Z is homothetic to
cos 9cp + sin 9(p, an associated minimal immersion of /, since / can be
written as
f= -[(l+cos9)(p+sin9(p]
= /2[cos9(p+sin§(p],
for some angle 9 and constant h.
In case G^ and G-< are congruent, by Theorem 1 we have that Z
is trivial, that is, Z = Bf + v where B is a skew-symmetric matrix. We
want to show now that either B is a multiple of a fixed complex
structure /, or that it is zero. If it is zero we are done. If it is
proportional to a complex structure J then we have by (5) that for
any tangent vector X and normal vector T|,
c<J(^r»,r|> = <V^Z,TI> =0,
showing that TM is invariant by J and consequently holomorphic.
Let us show now that B2 is a negative multiple of the identity
matrix. First observe that condition (5) becomes
<^,n> = o ,
946 M. DAJCZER AND L. RODRIGUEZ
which implies that B, and thus B2, leaves invariant the tangent and
normal spaces of the immersion /. Using the fact that B is constant
we obtain by the Gauss formula that
(6) VyBX + a(Y,BX) = VyBX = B(VyX)
=^(V^+2?(oc(Z,y)).
For a normal direction T| the T|-component of formula (6) gives us
(BX^Y) = <5(a(y,D),Ti>,
which is equivalent to
<z,^y> = <^w
because B is skew-symmetric. Since this is true for all X, Ye TM we
have that
A^ = BA^.
Using that A^ is symmetric, we have that
BA^ = (BA^ = A^f = - A^B
and
A^B2- B^.
Since both A^ and B2 are symmetric and commute they have the same
eigenvectors.
Because B preserves the tangent planes, observe that formula (6)
gives us that the tensor B restricted to TM is parallel, and thus also
B\ i.e.,
Vy^ == B\^yX)
for all X, Ye TM. Therefore all the eigenspaces of the symmetric matrix
B2 are parallel and since M"' is locally irreducible, B2 is a multiple of
the identity. If it is a positive multiple then we would have that B is
also, which is a contradiction since it is skew-symmetric. Thus B must
be proportional to a complex structure. D
Remark. — Of course the condition of locally irreducible is necessary
since otherwise we could use a different homothety on each factor.
INFINITESIMAL RIGIDITY 947
As a corollary we get a result on pairs of immersions (not necessarily
isometric) which make a right angle at every point, i.e., which satisfy
for each tangent vector XeTM,
(7) </^,^^>=0.
THEOREM 5. — Let /, g : M71 -> Rn+p be t\vo immersions \vhich have
the same Gauss map and \vhich make a right angle. Assume that f is
substantial and locally irreducible. Then M is a Kaehler manifold mth
the metric induced by /, and either f is holomorphic and g = cJf + v
for some complex structure J of IR^, constant c and vector i;, or f is
minimal and g = cj + v where J is the conjugate minimal immersion of
/.
Proof. — Since g^X = Xg = V^, then condition (7) implies condi-
tion (1) for Z = g . Since they have the same Gauss map then we have
also that condition (5) is satisfied and we can apply the previous
theorem to conclude the proof since Z = g cannot be a constant
vector. D
3. Families of deformable immersions.
Let F be an isometric variation of the isometric immersion /:
Af1 -> ̂ . In this case we can make use of the theory of infinitesimal
isometric variations to produce new immersions of At" into (R^, with
different induced metrics, which are also not isometrically rigid. We will
make use of the following.
PROPOSITION 6. — Assume that the isometric variation F has trivial
/3 n
variation vector field Z< = —for all t e l . Then F{ is trivial.
dt
Proof. — By assumption Z^ = Ef^ + v^ for all t e l . For any pair
of points x, y e M, we have
^-Wx) - F,(y)\\2 = 2<F,(x)-F,00,Z<(x)-Z,(30>
dt
= 2<^(x)-F,GO, B^F,(x)-F,(y))Y = 0.
Consequently, \\F,(x) - F,(y)\\ = \\F,(x) - F,(y)\\, for all t e l , and x,
y e M . D
948 M. DAJCZER AND L. RODRIGUEZ
THEOREM 7. — Let f: M" -> R^ be a deformable isometric immersion.
Then there exists a t\vo-parameter family of immersions F(t,s,x) :
J x R+ x AT -> ̂  which are not rigid.
Proof. — Let F(t,x): I x AT -^ ^N be a non-trivial variation of /.
Now by Proposition 6 there exists an interval J a I such that the
3Fvariation field Z^ == — is not trivial for all t e J . Defineot
F(t,s,x) = F(t,x) + sZ,(x).
As in Theorem 1, we conclude that these immersions are not rigid.
D
Observe that the new metrics induced by the F(t,s,x) are different
from the old one because, for any Ye TM such that VyZ< ^ 0, we
have
IIF^II2^ IIAT+^IIV^ZJI^ HAT.
We now consider the special case N == n + 1, and show that the
new metrics are quite different from the original ones.
PROPOSITION 8. — Let Z be a non-trivial variation vector field of an
isometric variation of a hy per surf ace f : M" -> Rn+l. If the metric induced
by the immersion F == f + sZ, s ^ 0 is conformal to the metric induced
by f, then f is a minimal real Kaehler hypersurface.
Proof. — If both metrics are conformal there exists a smooth
function (() ^ 0 such that
HV^ZH^ (|)(x)||Z||2,
for all X e TM. For any open connected subset U <= M where ^ > 0
we have that / and Z are two conformal immersions which make a
right angle. It follows from the theorem of [DR2] that / is a minimal
real Kaehler hypersurface and that Z is conformal to the associate
minimal hypersurface by a constant conformal factor, i.e., (() is constant
on U. The result follows. . D
Recall that real Kaehler hypersurfaces have been classified in [DG].
INFINITESIMAL RIGIDITY 949
BIBLIOGRAPHY
[A] C. B. ALLENDOERFER, Rigidity for spaces of class greater than one,
Amer. J. Math., 61 (1939), 633-644.
[CD] M. do CARMO and M. DAJCZER, Conformal rigidity, Amer J of Math
109 (1987), 963-985.
[DG] M. DAJCZER and D. GROMOLL, Real Kaehler submanifolds and uniqueness
of the Gauss map, J. Diff. Geometry, 22 (1985), 13-28.
[DR1] M. DAJCZER and L. RODRIGUEZ, Rigidity of real Kaehler submanifolds,
Duke Math. J., 53 (1986), 211-220.
[DR2] M. DAJCZER and L. RODRIGUEZ, Hypersurfaces which make a constant
angle, in «Differential Geometry», Longman Sc. & Tech Harlow
1990.
[GR] R. A. GOLDSTEIN and P. J. RYAN, Infinitesimal rigidity of submanifolds,
J. Diff. Geometry, 10 (1975), 49-60.
[S] R. SACKSTEDER, On hypersurfaces with no negative sectional curvature,
Amer. J. Math., 82 (1960), 609-630.
[Y] K. YANO, Infinitesimal variations of submanifolds, Kodai Math. J., 1
(1978), 30-44.
Manuscrit recu Ie 6 fevrier 1990.
M. DAJCZER & L. RODRIGUEZ,
I.M.P.A.
Estrada Dona Castorina 110
22460 Rio de Janeiro (Bresil).


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