ON RIGIDITY OF HARMONIC MAPPINGSINTO SPHERES

GABOR TOTH

1. Introduction

A map f: M -> M' of a compact oriented Riemannian manifold M into af

complete Riemannian manifold M' is harmonic if its energy E{f) = j II/JI2 vol (M)

M

is stable to first order with respect to variations of / [2]. Though, by the work ofT. Sunada [14], harmonic maps into nonpositively curved codomains are globallyrigid, an essential obstruction to proving rigidity in the case when RiemM ^ 0 is thelack of Hartman's uniqueness [3]. On the other hand it has long been noticed that,in all cases, harmonic maps behave nicely with respect to infinitesimal deformationspreserving harmonicity up to second order, that is, the second variation formula[8, 13] and Jacobi fields along harmonic maps have been proved to be useful inshowing rigidity of harmonic maps (see [2, 6, 9, 10, 11, 12, 13]).

The purpose of this paper is to study harmonic maps into spheres with variousrigidity properties when higher order terms of the expansion of the energy functionalalong a variation are also taken into account. In Section 2 we define the (geodesic)variations of a harmonic map given by translating the map along geodesies of aprescribed Jacobi field along this map. The concept of harmonic variation isintroduced [15] and its close relationship with the Jacobi fields is indicated(Theorem 1). In Section 3 infinitesimal and local rigidity of harmonic maps arestudied in detail; for example, by reducing the problem to that of the linear algebra,we show that harmonic embeddings / : Sm -*• S" with energy density e(f) = w/2 arerigid (Theorem 2). (For examples of nonrigid harmonic embeddings, see [17].) InSection 4 certain metric spaces of locally rigid harmonic maps are introduced andtheir classification is reduced to that of the canonical inclusion map i: Sm -• S".

Throughout this note all manifolds, maps, bundles, etc. will be smooth, that is, ofclass C°°. The report [2] is our general reference, adopting the sign conventions of[5].

We thank Prof. J. Eells and Prof. A. Lichnerowicz for their valuable suggestionsand encouragement at the conference on invariant metrics, harmonic mappings andrelated topics held in Rome in 1981. We also thank Prof. A. Lee for usefuldiscussions on the matrix calculus used in this paper and for giving a proof ofLemma 2.

2. Jacobi fields and harmonic variations

Let / : M -• S" be a map of a compact oriented Riemannian manifold M ofdimension m into the Euclidean n-sphere Sn. A vector field v along / , that is, a sectionof the pull-back bundle F = f*(T(S")), gives rise to a (geodesic, 1-parameter)

Received 9 November, 1981.

J. London Math. Soc. (2), 26 (1982), 475-486

476 GABOR TOTH

variation t -> / , = exp o {tv), teU. Then U G C ° ° ( F ) is said to be a harmonicvariation if the maps ft:M-*S" are harmonic [2] for all teU. The set of allharmonic variations (or the variation space) of the given map / is denoted byV(f) c C°(F).

If t -> /,, t G U, is an arbitrary (not necessarily geodesic) variation of / through8f.

harmonic maps then v = —ot

the Jacobi equation

is a Jacobi fieid along / [2], that is, v is a solution oft = 0

j f V = - V2 v + trace R{f^,v)f^ = 0 ,

where V is the induced connection of the bundle <!; = F ® A*(T*(M)); V2u isthe trace of the bilinear form {X, Y) -> VxVyy-VV x yu (see [11]); /„, G C°°(^) is thedifferential of / and R denotes the Riemannian curvature tensor of S". Thedifferential operator Jf e Diff2 {F, F) is strongly elliptic [11] over the compactmanifold M, and hence solutions of the Jacobi equation have uniqueness in theCauchy problem; the nullity Null (/) = dim J(f), J{f) = ker Jf, and theMorse index of / (that is, card {eigenvalues (Jf) < 0}) are finite. By a recenttheorem of P. F. Leung [6] the index of a nonconstant harmonic map / : M -> S",with n ^ 3, is strictly positive; by an earlier result of R. T. Smith,index(ids,,) = n+l [13]. For the nullity of idsn see [13] and Section 3 below.

THEOREM 1. Let v be a Jacobi field along the harmonic map f : M -> S" and write

T(v) = {t G U | ft = exp o {tv) is harmonic}.

Then the following cases can occur:

(1) T(v) consists of at most two points (including 0) and \\v\\ is not constant,

(2) \\v\\ is a non-zero constant and T(v) = - — Z ,INI

(3) T(v) = U, that is, v e V(f), and this is the case if and only if\\v\\ is constantand trace </„,, Vy> = 0.

Proof. Given 0 =•£ v e C°°(F) the induced variation

t > (/,)* e C » ( F ® T*(M)), F = (/r)*(T(S")),

can be conveniently described by considering the 1-forms

e CW(F ® T*(M))

with values in F, where T\'»[V] : F' -> F'" is the isomorphism defined by the paralleltransport along the geodesic segments t -> /,(x), r G \t', t"~\ (or t G [t", t']), x G M. Forsimplicity, we write T{'. = t'r[i;] ® irfA*(T.(M)) and omit 0 in / 0 , T'O, etc. Thent -• (./;)»*,, with X , G TX(M), x G M, is a Jacobi field along t -> /,(x), t e R, andapplying T' to both sides of the corresponding Jacobi equation and omitting Xx we

ON RIGIDITY OF HARMONIC MAPPINGS INTO SPHERES 477

obtain

(A) ^ 2 .

with initial data

(B) PM = f* and( = 0

(cf. [15, 16]). Using the well-known formula [5, Vol. I, p. 203], we have

R{X,Y)Z = (Y,Z}X-(X,Z)Y, X,Y,ZeX{Sn);

the initial value problem (A) with (B) can be solved explicitly

/ ^

\ IMIINI V \ IMI/IM

(in the case when vx = 0 we take Py(0 = /* + tVv at x e M). In order to compute thetension field t(/t) = div (/,)„, e C°°(F') [2] in terms of /„, and y we need the identity

( T ' O V ' O T , ) W - V W = - i? ( PB(s)ds,ujw, weC°°(F),o

relating the induced connections V and V on g = Ff ® A*(T*(M)) and ^,respectively. (This can be shown directly by using local symmetricity of S" and thedefinition of R by parallel transport; see also [15].) Then, as in [16], we get

T'(T(/,)) = T* trace {V'(/,U = trace {(t< oV< o xt)Pv(t)}

= - (v*Pv(t) + trace

in particular, ft: M -*• S" is harmonic if and only if

(C) ¥(», 0 = V*PB(t) + trace | n f P^s)^, ^ PB(t)| = 0 .

o

On substituting the solution Pv(t) of (A), (B) into (C) an admittedly long calculation

478 GABOR T6TH

yields

trace {*</„„)/.}

, ^

^

t r a c e

(in the case when vx = 0 we take the corresponding limits). Assume now that/ : M -> Sn is harmonic and v e J(f). By Cauchy uniqueness, the complement M' ofthe zero locus of ||u|| (not identically zero) is dense (and open) in M. Let 0 j= t e T{v)be fixed. Using the Jacobi equation and the explicit formula for *¥{v, t) above, aroutine calculation shows that the equation

l-cos(2r|M|) , /r ,„ x

, trace<R(L, v)Vv, u>Hull

—— {II^H2<V2y, u>-trace </?(Vu, y)Vu, u>} = 0

holds on Af'. If 0 ^ f' e T(u) we get

1 — cos(2f||u||) 1—cos(2r'||i;|

sin(2t||w||)-2r|N|y|| trace(R(L, v)Vv, v) = 0 (on M').

Write U cz M' for the (possibly empty) complement of the zero locus oftrace </?(/*, v)S7v, v). Then

) l-cos(2t'||y||)

ON RIGIDITY OF HARMONIC MAPPINGS INTO SPHERES 479

is valid on U. Assuming that \\v\U\\ is non-constant, we can choose a non-emptyopen interval / <= im(||2u|t/||) c= R+ such that

l -cos(d) I - cos (t'k)

sin (tk)-tk sin (t'k)-t'k

for all k 6 /. Both sides of this equation are analytic in k and thus the equality holdsfor all k ^ 0. Comparison of the least positive roots gives t = t', that is, T(v) consistsof at most two points {0, t} and we have case (1) of the theorem. If, on the otherhand, \\v\U\\ is constant, then, in particular, V2(||i;||2) ^ 0 on U. Now the formula for<¥(u, t), v} above shows that x e M' — U implies that

0 = ||uJ|2<V2u, v)x-trace(R{S7v,v)Vv,v}x

= ilNI2(V2(IMI2))*-trace <Vu, u>2 ,

using the formula for R above, and so V2(||i;||2) ^ 0 on M' — U. Since M' c M isdense we get V2(||u||2) H on M and, by Hopf's lemma [5, Vol. II, p. 338],compactness of M implies that ||u|| is constant. Without loss of generality we mayassume that ||i;|| = 1. Since the antipodal map of S" is an isometry, nZ <= T(v) isalways satisfied. Thus either T(v) = nZ and we have case (2), or there existst0 e (0, n) n T{v). Then

OF(Mo),»> = (cos(2t0)-l)trace</;,Vi>> = 0

and so trace </„., Vu> = 0. Using this and the fact that ||u|| is constant, one can easilycheck that *¥(v, t) = 0 for all t e U, that is, v e V(f) and we have case (3). The proofis complete.

REMARK 1. If v e J{f) and ft — exp o (tv), t e U, is harmonic in the direction xtv,that is, if <r(/f), iry> = 0, then veV(f). Indeed, the hypothesis implies that

iv, t), v) = 0 and we can argue as above.

REMARK 2. The Euler class of the tangent bundle of an even sphere is twice thegenerator which, in the case when m = n and is even, yields that a harmonic mapf: M -*• S" with nontrivial variation space V(f) must have Brouwer degree zero.

REMARK 3. The energy £(/,) is constant along a harmonic variationt -> ft = exp o (tv), v e V(f), since ft is a critical point of E for all teU. In fact, amore precise statement appears in [16], namely, that v e V(f) if and only if v e J(f)and the energy density e(ft) = \trace ||(/,)J|2, as a scalar on M, is the same for allteR.

3. Rigidity

Let / : M -*• S" be a harmonic map and put

480 GABOR T6TH

Then K(f) e J(f) is a linear subspace and, by Theorem 1,

V(f) = {ve K(f) | ||u|| is constant}.

In what follows we compute K(i), where i: Sm -> Sn, m ^ n, is the inclusion map.By S" c M" + l, vectors tangent to 5" are identified with their translates at the origin.Then the canonical base vectors em + 2,...,en + leU" + l define k (= n — m)orthonormal parallel sections Wl,..., Wk of the normal bundle of i. Assuming thatt; e K(i), the (orthogonal) decomposition

k

v = w+ £ <y, Wj)Wj, We

splits the Jacobi equation Jfv = 0 into the system

j j} = 0 , j

= 0

(cf. [16]). It follows that since the scalar (v, WJ) is an eigenfunction of the Laplacianon Sm, it is the restriction of a homogeneous linear function on [Rm + 1 [1], that is,there exists a unique vector b/elRm + 1 with (vx, WJ

X) = (bj,x), xeSm. Moreover,denoting by /? the 1-form on Sm that corresponds to W by duality, the last equationcan be rewritten as

AjS-2(m-l)j3 = 0 .

On the other hand, 0 — trace <!„,, Wv} = trace (i^, VW) = — V*/? and a result ofA. Lichnerowicz [7, Proposition 1, p. 80], shows that W€so(m + l) is a Killingvector field. The converse is obvious; thus we obtained the following.

LEMMA 1. For the canonical embedding i:Sm^> S" the vector field v along ibelongs to K(i) if and only if the tangential part W of v is a Killing vector field andthere exist vectors bx, ...,bke Um + 1, k = n — m, such that

vx= Wx+ X <bj,x>W{, xcSm,

where {W1,..., Wk) is the canonical orthonormal system of parallel sections of the

normal bundle of i.

In particular, dimK(i) = —^-—- +(n-m)(m + l) and K(ids,,) = so{n + l). The

last equality, in the case when n ^ 2, follows from a sharper result of R. T. Smith[13], in which he showed that Null(ids,,) = n(n + l)/2 or, equivalently, thatJ(ids,,) = so{n + \), if n ± 2. Note, however, that, by the presence of conformaldiffeomorphisms, dimJ(idS2) = 6 (cf. [12]). Furthermore, using Lemma 1, we havethe following.

PROPOSITION. / / / : M -> S" is a harmonic Riemannian submersion and v € K(f) is

ON RIGIDITY OF HARMONIC MAPPINGS INTO SPHERES 481

projectable, that is, if f(x) = f{x') implies that vx = vx., x, x' e M , then v = X of,

where X e so(n + 1). //, moreover, v e V(f) then ft = exp o {tv) = (f)to f, teU, where

is the l-parameter group of isometries induced by X.

Proof. As v takes its values in T(S") there is a unique vector field X on thecodomain S" such that v = X o / is valid. By choosing suitable local orthonormalframes in M and in Sn we obtain V2u = V2(X o / ) = (V2X) o / (cf. [17, Proposition2]) and hence

(V2X)of = trace/*(/«,, v)& =

Similarly, trace </„., Vu> = trace <(idsn)#, VX} o / . These two equations imply thatX e K(ids,,) and so, applying Lemma 1 in the case when m = n, we get X e so(n +1).If u e K(/) then H H is constant and thus for any vector field Y on S" we have

(Y,VxXy = -(X, VyX> = - |Y | |X | | 2 = 0

(cf. [5]). It follows that S7XX = 0 on S" or equivalently that every integral curvet —• 4>((x), t e R, x e S", of X is a geodesic. Hence

_/;(*) = exp (tvx) = exp (t-Y/U)) = 0,(/(x))

for all x e M ; this completes the proof.

A harmonic map f: M -> S" is said to be infinitesimally rigid if for everyprojectable veK(f) there exists Xeso(n + \) such that the equation v = Xofholds. (Note that X o f e K(f) is always satisfied for X e so(n +1)). The map / islocally rigid if for every projectable harmonic variation v there exists a l-parametersubgroup {(j)t) c= O(n +1) such that f = exp o {tv) = 0, o / for all t e U. (Obviously,isometries of the codomain preserve infinitesimal and local rigidity.) The reason forthe use of K(f) to denote infinitesimal rigidity is that the set of all infinitesimalisometries of S" precomposed by / forms a linear space which suggests a choice ofK(f) defined by linear constraints. As for local rigidity, for projectablev = X o / e K(f), we cannot expect exp o (tX o / ) = 0, o / to be valid for allt e IR, where (4>,) denotes the l-parameter group of isometries induced by X. Hence,in the definition of local rigidity, we have to restrict ourselves to the subsetV(f) <= K(f) (which is not a linear subspace in general, cf. [17]). Note that in all theknown examples, infinitesimal and local rigidities are equivalent. By the aboveproposition, any harmonic Riemannian submersion is infinitesimally and locallyrigid.

THEOREM 2. The canonical inclusion map i: Sm -> S" is infinitesimally and locallyrigid.

Proof. Let v e K(i) be fixed and consider the decomposition of v in Lemma 1.For p, q G N, let M(p, q) be the vector space of (p x g)-matrices; M(p, q) = L{W, Up).Finally, write BeM{k,m + \), k = n — m, for the matrix whose rows areb!,..., bk G Um + 1, and which occurs in the decomposition of v. By identifying thetangent vectors of S"e!R" + 1 with their translates at the origin, the vector field valong / gives rise to a matrix Ae M(n + l,m+l) with upper block W e so(m+l)

482 GABOR T6TH

and lower block B e M(k, m + 1) such that vx = Ax holds for all x e Sm. If Y e so{k) isany skew-symmetric matrix then

~w(D) " " W 41is a Killing vector field on S" with v = X oi and hence i:Sm^> S" is infinitesimallyrigid.

Now let v e V{i) with ||t;|| = 1. Then, for x e Sm, we have

1 = ||t;||2 = \\Ax\\2 = (ATAx,x} = ({WTW + BTB)x,x},

that is, we get WTW + BTB = Im + l (the identity). (In particular, by the identificationK{i) = so(m + \)xM(k, m + 1) the variation space of i has the formV{i) = U{{W, B) 6 K(i) | - W2 + BTB = Im + 1}.) To prove local rigidity, we have tofind Y e so(k) such that the corresponding Killing vector field X above has theproperty that the integral curve t -> <j)t{x) = e'xx of X is a geodesic for all x e Sm.

This holds if and only if —-jetXx = —etXx, that is, if and only if X2x = — x for all

x G Sm. By computing X2 in terms of the blocks of X and using the fact that- W2 + BTB = Im +!, we have to choose Y e so(/c) so as to satisfy the equality

Let ey,..., em+l e Um + i be the canonical base vectors and let c l 5 . . . , cm + 1 e IRk denotethe columns of B. Having in mind that the above equation must be valid we defineYCJ = —BWej,j = l , . . . , m + l . Then Y is skew on the vectors cl,...,cm + l since

<yc,-,c,> = -(BWej,Bej.) = -(WepBTBer) = -

2 + Im+l)er> = <ej,(W2 + Im+1

Putting V = span{c l 5 . . . , c m + i } c Um + l we define

where x = £ o^c, e F. We have

m+1

<Yx,C/>= (

m+1= -<x ,Y C / >

ON RIGIDITY OF HARMONIC MAPPINGS INTO SPHERES 483

and so Y is well defined (that is, Yx does not depend on the particulardecomposition of x) and skew on V. Extending Y\V to a skew-symmetric linearendomorphism Y eso{m + l) the equation BW+YB = 0 holds; this completes theproof.

REMARK 1. An immediate consequence of Theorem 2 is that any harmonicRiemannian submersion / : M -> S" onto a totally geodesic submanifold of S" isinfinitesimally and locally rigid.

Indeed, assuming that im / = Sm c S", any projectable vector field v along / canbe written uniquely as v = w o / , where w is a vector field along the inclusioni: Sm -> S". If, moreover, v e K(f) then by choosing suitable local orthonormalframes a similar computation as in the proof of Proposition 2 in [17] shows that

V2y = ( V 2 w ) o / = trace/?(/„, v)f+ = (trace KO*, w)ij of

and trace </„,, Vu> = (trace (i^, Vw» o / , that is, w e X(j). Thus Theorem 2 impliesthe existence of a vector field Areso(n + 1) with Xoi = w and it follows thatX o / = v, which completes the proof in the infinitesimal case. As for local rigidity, ifv e V(f) is as above then w e V(i) and, again by Theorem 2, we can chooseX Gso{n + 1) so as to satisfy VXX oi = 0. Thus VXX of = 0 which implies localrigidity of / .

REMARK 2. From the proof of Theorem 2 we see thatV(idsn) = so{n + l)n USO{n + l), in particular, V(idsn) = {0} for n even. (Thisfollows also from Remark 2 after Theorem 1.)

For odd spheres a stronger version of Theorem 2 is valid. First we state thefollowing lemma, due to A. Lee.

LEMMA 2. Let Weso(m+l) and BeM(k,m + l) with -W2 + BTB = Im+l. Ifn = k + m is odd then the equation BW+YB = 0 has a solution Y such that thecorresponding matrix X in (D) belongs to so{n+1) n O(n +1).

Proof. We verify the statement for k ^ m +1 and m = 2r — 1, r E N. (The othercases can be treated similarly.) By the singular values decomposition of matrices,there exist V e O(k) and U e O(m +1) such that B = VY.UT, where

S = eM(/c,m +

and px ^ ... ^ p2 r . (The numbers pj, j = l , . . . ,2r, are often called the singularvalues of B.) Then BTB = l /diag(p2, . . . , pl,)UT and using the equation-W2 + BTB = Im + 1 we have -W2 = t / d i a g ( l - p 2 , . . . , l-p2

2r)UT. On the other

484 GABOR T6TH

hand, U diagonalizes WTW and hence, by an appropriate choice of U, we haveW = UAUT where

Thus, computing VK2 = UA2UT, we get p2j- i = P2j> t n a t is> ^ +°"j = 1> where°i = PIP 7 = !>•••>>*. Because k — (m + \) = k — 2r is even we can defineY = V(A 0 £) KT e so(fc), where

i] [-1 iThen BW+YB = V(LA — ET.)UT = 0 and a staightforward computation showsthat the matrix X in (D) is orthogonal; this completes the proof.

THEOREM 3. / / n is odd and v is a harmonic variation of the canonical inclusionmap i: Sm -> S" then there exists a Killing vector field X e so(n +1) of constant normon S" such that v = X o i.

Proof. Keeping the notation of the proof of Theorem 2, let v e V(i), where ||u|| isa constant, say 1, and write Weso(m + l) and B e M(k,m + l) for the matricesdefined by v. Choosing Y e so(k) as in Lemma 2 we have v = X oi and\\XX\\2 = HXxll2 = ||x||2 = 1 for all x e Sn. Thus the theorem is proved.

REMARK. Theorem 2 can also be proved by using the singular valuesdecomposition; however, we preferred to give an elementary and direct constructionfor a solution of the equation BW + YB = 0 in so{k).

Returning to the general situation, let / : M -» S" be an infinitesimally rigidharmonic map and assume that i m / c So = exp(/!(!(TX0(M)) for some xoeM. Ifv e V(f) is projectable then infinitesimal rigidity of / implies the existence of aKilling vector field Xeso(n + l) with v = X of. Moreover, as HXo/H = ||i;|| isconstant it follows that \\X\\ is constant on a relatively open neighbourhood of x0 inSo. Thus HA Soll is constant, that is, X oi0€ V(i0), where io:So -» S" is the canonicalinclusion. By local rigidity of i0 there exists X e so{n + 1) such that X o i0 = X o i0

and the integral curves t -+ $r(x), t e U, of X are geodesies for all x e So. Hence / islocally rigid and we have proved the following.

THEOREM 4. Let f: M —*• S" be an infinitesimally rigid harmonic map withi m / c= exp(/,,.(TY0(M)))/or some x0 e M. Then f is locally rigid.

The condition for the image of / is satisfied if either / has maximal rank, withsurjective /,., at some x0 e M, or / is totally geodesic. (In the latter case / mapsgeodesies onto geodesies and a straightforward argument shows that i m / c S" is atotally geodesic submanifold.)

4. Metric spaces of locally rigid harmonic embeddings

Write Harm (M, S") for the set of all harmonic maps f: M -> S". Two maps /

ON RIGIDITY OF HARMONIC MAPPINGS INTO SPHERES 485

and / ' in Harm (M,S") are called equivalent, / ~ / ' , if there exist maps/ 0 , . . . , / k + 1 E Harm (M,Sn) and vectors v> e V(fj), j = O,...,k, such thatf° = / , fk + 1 = / ' and expo?/ = fj+l,j = 0,...,/c. Clearly ~ is an equivalence

k

relation. If / ~ / ' , write p( / , f) for the infimum of the numbers £ ||iy*|| for which

v>eV{f>\ expovi = fJ+1, j = O,...,k and f° = / , fk+1 = / \ ° T h u s p is adistance function on the equivalence class N{f) <= Harm (M,S") containing theharmonic map / : M -> 5".

Our present aim is to determine to what extent N(f) depends on the particularchoice of / . In this way, for locally rigid harmonic embeddings / : M -*• S" (whereV(f') is the same for all / ' e N(f)), the problem of describing N(f) is reduced to thatof describing N(i), where i: Sm -> S" is the canonical inclusion map for some meN.

Let / : M -> S" be a map and define the span of / by S0(f) = S" n span im / ,where i m / is considered as a subset of U" + 1 (=> S").

THEOREM 5. Let f: M -> S" be a locally rigid harmonic embedding withm-dimensional span. Then there exists a bisection *¥ : V(f) -» V(i) preserving theadditive relations, where i: Sm —> S" is the inclusion map. Moreover the metric spacesN(f) and N(i) are isometric.

Proof. We may suppose that the span of / is Sm c S". If v e V(f) then localrigidity of / implies the existence of a Killing vector field X e so(n+1) such that wehave v = Xof and (VxX)of = 0. Denote by F that (arcwise) connectedcomponent of the zero locus of VXX which contains im/ . As in the proof ofTheorem 2, F consists of eigenvectors of X2, that is, X2x = — \\Xx\\2x for all x e F.The eigenvalues of X2 form a discrete set and hence H H is a constant, say c, on Fand F c S" on a totally geodesic submanifold, since it is the intersection of the linearsubspace k e r ( X 2 - c 2 / m + 1) c r + 1 with Sm a F; thus we obtain that X oie V(i).

IfXeso{n + l) with v = Xof and ( V ^ X ) o / = 0 then (X-X)of = 0. Sincethe connected components of the zero locus of a Killing vector field are totallygeodesic submanifolds [4, Theorem 5.4, p. 62], it follows that (X-X) | Sm = 0, thatis, X o i = X o i. Define *¥ : V(f) -• V(i) by ¥(u) = I o i , where X e so(w +1) withXof=v and (VxX)of= 0. By the above reasoning, T is well-defined and injectiveand it preserves the additive relations. Local rigidity of the inclusion map i: Sm -> S"implies that *¥ is surjective, which proves the first statement. As for the second, defineO : N{f) -> N{i) such that, for / ' e N(f), O(/ ') is the inclusion of the span S0(f) of/ ' into 5". A straightforward argument shows that O maps N(f) onto N(i)isometrically; this completes the proof.

REMARK. If / : T2 -*• S3 is a harmonic embedding with e(f) = j then^o(/) = ^3- An easy computation shows that the geometric lattice of idsi has theform [17]

G(ids>)

486 ON RIGIDITY OF HARMONIC MAPPINGS INTO SPHERES

and, by [17], there is no bijection *¥: V(f) -*• V(idsi) preserving the additiverelations. Again, we obtain that / is not locally rigid.

COROLLARY. Let f: M -> S" be a locally rigid harmonic embedding with n odd.Then there exists a contractive surjection 6 : N(idsn) -> N(f).

Proof. By the previous theorem it is enough to consider the case when/ = i:Sm -+ S". F o r / ' e N(idsn), def ine 9{f) = f ' o i e N(i). C l e a r l y , t h e m a p0: N(idsn) -*• N(i) is contractive. By Theorem 3 any v e V(i) can be extended to aharmonic variation of idsn which implies that 6 is surjective.

References

1. M. BERGER, P. GAUDUCHON and E. MAZET, Le spectre d'une variete Riemannienne, Lecture Notes inMathematics 194 (Springer, Berlin, 1971).

2. J. EELLS and L. LEMAIRE, 'A report on harmonic maps', Bull. London Math. Soc, 10 (1978), 1-68.3. P. HARTMAN, 'On homotopic harmonic maps', Canad. J. Math., 19 (1976), 673-687.4. S. KOBAYASHI, Transformation groups in differential geometry, Ergebnisse der Mathematik 70

(Springer, Berlin, 1972).5. S. KOBAYASHI and K.NOMIZU, Foundations of differential geometry, Vols. I—II (John Wiley & Sons,

New York, 1963, 1969).6. P. F. LEUNG, 'On the stability of harmonic maps', preprint, University of Notre Dame, 1981.7. A. LICHNEROVICZ, 'Varietes Kahleriennes a premiere classe de Chern non negative et varietes

Riemanniennes a courbure Ricci generalises non negative', J. Differential Geom., 6 (1971), 47-94.8. E. MAZET, 'La formule de la variation seconde de Penergie au voisinage d'une application

harmonique', J. Differential Geom., 8 (1973), 279-296.9. O. NOUHAUD, 'Deformationes infinitesimales harmoniques', Rend. Mat. (6), 10 (1977), 87-108.

10. R. SCHOEN and S. T. YAU, 'Compact group actions and the topology of manifolds with non-positivecurvature', To.pology, 18 (1979), 361-380.

11. J. SIMONS, 'Minimal varieties in Riemannian manifolds', Ann. of Math., 88 (1968), 62-105.12. R. T. SMITH, 'Harmonic mappings of spheres', Thesis, Warwick University, 1972.13. R. T. SMITH, 'The second variation formula for harmonic mappings', Proc. Amer. Math. Soc, 47

(1975), 229-236.14. T. SUNADA, 'Rigidity of certain harmonic mappings', Invent. Math., 51 (1979), 297-307.15. G. TOTH, 'On variations of harmonic maps into spaces of constant curvature', Annali di Matematica

(4), 127 (1982), 389-399.16. G. TOTH, 'On harmonic maps into locally symmetric Riemannian manifolds', Conference an invariant

metrics, harmonic mappings and related topics, Symposia Mathematica 26 (Instituto Nazionale diAlta Matematica, Roma, 1982).

17. G. T6TH, 'Construction des applications harmoniques non rigides d'un tore dans las sphere', Annals ofglobal analysis and geometry, to appear.

Mathematical Institute of the Hungarian Academy of Sciences,H-1053 Budapest,

Realtanoda u. 13-15,Hungary.