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Proc. Natl. Acad. Sci. USA Vol. 82, pp. 2217-2219, April 1985

Harmonic of S2 into a complex Grassmann (harmonic sequences/Frenet harmonic sequences/fundamental collineations/harmonic flags/"crossing" and "turning" constructions) SHIING-SHEN CHERNt AND JON WOLFSONt tDepartment of Mathematics, University of California at Berkeley, and Mathematical Sciences Research Institute, Berkeley, CA 94720; and tDepartment of Mathematics, Rice University, Houston, TX 77251 Contributed by Shiing-shen Chern, December 5, 1984

ABSTRACT Let G(k, n) be the Grassmann manifold of all the coefficients wAB- are the Maurer-Cartan forms ofthe uni- Ck in C., the complex of k and n, respec- tary U(n). They are skew-hermitian; i.e., tively, or, what is the same, the manifold of all projective spaces Pk-1 in P.-,, so that G(1, n) is the complex projective WA + (HA=° (OBA = (BAB [1.7] Pn-1 itself. We study harmonic maps of the two-dimen- sional S2 into G(k, n). The case k = 1 has been the Taking the exterior of Eq. 1.6, we get the Maurer- subject of investigation by several authors [see, for example, Cartan Din, A. M. & Zakrzewski, W. J. (1980) Nucl. Phys. B 174, 397-406; Eells, J. & Wood, J. C. (1983) Adv. Math. 49, 217- dwAj = WoAC A ccB. [1.8] 263; and Wolfson, J. G. Trans. Am. Math. Soc., in press]. The harmonic maps S2- G(2, 4) have been studied by Raman- athan These equations contain all the local of G(k, n). [Ramanathan, J. (1984) J. Differ. Geom. 19, 207-219]. An element Ck of G(k, n) can be defined by the We shall describe all harmonic maps S2- G(2, n). The meth- Z1 A ... A 7 defined to a od is based on several geometrical constructions, which lead Zk 0, up factor. This defines a G- from a structure on G(k, n), with G = U(k) x U(n - k). (We have given harmonic to new harmonic maps, in which called such a structure a Segre structure.) In particular, the the projective spaces are related by "fundamental col- form lineations." The key result is the degeneracy of some funda- mental collineations, which is a global consequence, following from the fact that the domain manifold is S2. The method ex- ds2 = (alm, wCi [1.9] tends to G(k, n). is a positive definite hermitain form on G(k, n) and defines Geometry of G(k, n) an hermitian . Its Kahler form is i We consider C, equipped with the standard hermitian inner fl= w.,, A w~r. [1.10] product. That is, for Z, W E C,

1.8 it can be W = (w1, ..., [1.1] By using Eq. immediately verified that il is WJ, closed, so that the metric ds2 is kahlerian. we have By the expressions for dw,,t7 we see that the forms are w,,j,coo; their exterior give the curva- ture forms of the metric dS2. (Z, W) = ZAWA = E ZAWX. [1.2] Surfaces in G(K, n) Throughout this note we will agree on the following ranges of Consider an oriented M immersed by a smooth map indices: f into G(k, n). It acquires an induced riemannian metric and hence a complex structure. Using the latter, we write the 1-A, B, C, ...-.n, 1 a 3, y, ...k, induced metric as ds 2 = sp5, sp being a complex-valued one- form, defined up to a factor of 1. For x E M k + 1 -'< ... _'4-' i, j, n, [1.3] the image f(x) E G(k, n) has an orthogonal space f(x)' E G(n - k, n), which describes a surface M'. If Z E f(x), then and we shall use the convention -ZA = zX and also the summa- tion convention. A frame consists of an ordered set of n vec- pX + mod tors ZA, such that dZA spY, f(x), [2.1] where X, Y E f(x)'. If Z E C, - {0}, we denote by [Z] the Zi A ... A zn,# O. [1.4] in P,-1 with Z as the homogeneous coordinate vector. Then It is called unitary, if .-+ .-+ a: lZI [XI I .Jzl WI [2.2] (ZA, ZB) = SNAB [1.5] define projective collineations of the projectivized space If we write [f(x)] into [f(x)']. We shall call these thefundamental col- lineations. Dually there are adjoint fundamental collinea- dZA = WOABZB, [1.6] tions from [f(x)'] to [f(x)]. To express the situation analytically we choose, locally, a The publication costs of this article were defrayed in part by page charge field of unitary frames ZA, so that Z(X) span f(x). Then payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact. f* Wacs = aaijP + b [2.3] 2217 Downloaded by guest on October 1, 2021 2218 Mathematics: Chern and Wolfson Proc. NatL Acad ScL USA 82 (1985) The fundamental collineations a and a send [Za] to [Xa] and quence are of the same , the sequence can be ex- [Ya], respectively, where tended in both directions, giving

Xa = a ,Zi, Ya = b -iZi. [2.4]

-" < The metric ds2 has a connection form p, which is a real ... <-- L-1 (a Lo* Li . .... [2.10] one-form satisfying the dip = -i p A (p. [2.5] A harmonic sequence such that any two members are or- thogonal will be called a Frenet harmonic sequence. A har- Taking the of Eq. 2.3 and using Eqs. monic sequence is called full if its members span the whole 1.8 and 2.5, we get Pn-1. An example of a Frenet harmonic sequence is given by the vertices ..., of the Frenet frame of a holomor- A (p + Dba, A 5 = 0, [2.6] Lo, L1, Ln-1 Dac, phic Lo(x), x E M, in G(1, n) = Pn-1. The Frenet for- can be expressed as a Frenet harmonic sequence where mula

' a 0 ) Da - = da - a#; way + a Zwj - i aap, Lo L. .... 0 Ln-l- [2.11] Dba7 = dba- - b7Wag + baj-ji + i bajp. [2.7] In fact, the fundamental theorem on harmonic maps S2 -3 Define P _1 is that any such map is a member of a Frenet harmonic sequence for then it can be obtained from a holomorphic DXa = dXa - (Wan + i p aTB)X3S, curve through fundamental collineations. For k = 2 our process can be briefly described as follows: DYa = dYa - (WaT - i p Sa-)YS1 [2.8] If the harmonic sequence degenerates, we are reduced to a harmonic map into P,,._. If it does not, we will "straighten" Then the condition for the map f to be harmonic (1) is given it into a full Frenet harmonic sequence. This is done through by Theorem 1. two constructions, which we call crossing and turning, re- THEOREM 1. The property that f is a harmonic map is giv- spectively. Their success depends on the vanishing of cer- en by one ofthefollowing conditions, which are equivalent: tain holomorphic differential forms and thus on the fact that the domain manifold is S2. Their inverse processes depend (i) Da, -0, mod 5p, on choices of holomorphic sections of Pl-bundles. (i) Dba7 a0, mod S5, Vanishing Theorems (iii) DX,- 0, mod Z1, Sp, The restriction on the harmonic maps whose domain mani- (iv) DYa 0, mod ZB, Si. fold is S2 arises from the fact that S2 has no holomorphic differential forms of positive degree except zero. From our From this criterion we immediately draw the conclusion that analytical data we are able to construct such forms and ob- a holomorphic or an anti-holomorphic map of M into G(k, n) tain in this way strong conclusions. Our first "vanishing the- is harmonic. Thus we shall study harmonic maps that are not orem" is Theorem 3. ± holomorphic. We have also the following theorem. THEOREM 3. Consider a harmonic map f:S2 -- G(k, n). Let THEOREM 2. Let f:M -* G(k, n) be a harmonic map. Then (i) f': M -. G(n - k, n) is harmonic, where f'(x) = f(x)', x E cat= [3.1] M. a.tb#. (ii) The images of [f(x)] under the fundamental collinea- Then tions 8, d are ofconstant dimensions, say k, - 1, k2 - 1, and the maps ofM into G(k1, n), G(k2, n) so defined are harmon- det(ca, + t 58T) = tk identically in t. [3.2] ic. Denote the images by a[f(x)], a[f(x)], respectively. If ki = k (resp k2 = k), then the image under a (resp a) of W[f(x)] This follows from the fact that, with t as a parameter, (resp a[f(x)]) is [f(x)] itself. det(caX- p2 + t 3,-#) is a holomorphic . This (iii) The kernels of the fundamental collineations a, a are theorem, in different but equivalent formulations, was of constant dimensions. If their orthogonal complements in known to Ramanathan (2), Uhlenbeck (personal communica- [f(x)] are ofdimensions 11 - 1, 12 - 1, respectively, the maps tion), and others. so defined into G(11, n) and G(12, n) are harmonic. Our next vanishing theorem is concerned with a Frenet It is advantageous to use projective geometry, and we harmonic sequence, as follows. write [f(x)] = Lo, being a of dimension k - THEOREM4. Let Li:S2 - G(k, n), 0 i-' _s - 1, n _ ks, be 1. Continuing the construction in part ii of Theorem 2, we get harmonic maps that form a Frenet harmonic sequence two sequences of harmonic maps Lo yL1 s... p~s-.a [3.3] [2.9] Let ir:L-.1 -l Lo be the standard . Then the map

connected by fundamental collineations. Each of the se- 7ro a:Ls-1 - Lo [3.4] quences in 2.9 will be called a harmonic sequence. By con- struction two consecutive projective spaces of a harmonic is degenerate. When n = ks, a:L,-1 - Lo, so in this case the sequence are orthogonal. If the members of a harmonic se- fundamental collineation is degenerate. Downloaded by guest on October 1, 2021 Mathematics: Chern and Wolfson Proc. NatL Acad Sci USA 82 (1985) 2219

The proof of this theorem again relies on the vanishing of a volves the choice of a holomorphic section of a Pl-bundle holomorphic differential. More vanishing theorems will be over S2-i.e., that of a meromorphic over S2. used. To explain turning consider a harmonic flag. Its harmonic point is a of the Frenet frame of some holomorphic Harmonic Maps of S2 into G(2, n) curve in P,,1. In our treatment harmonic flags always arise from degenerate fundamental collineations. Consider there- We shall describe our crossing and turning constructions. To fore the Frenet harmonic sequence express the situation vividly we shall call the image, under a harmonic map, of a point x E M in G(1, n) (resp G(2, n)) a x1 -l 4 [4.4] harmonic point (resp a harmonic ). Similarly, a harmonic xo Po, flag consists of a harmonic point with a harmonic line through it. The crossing construction passes from a harmon- where Po = ima and Pi = kera E Xo are points and (Pl, Xo) is ic line to a harmonic flag and the turning construction passes a harmonic flag. Denote by q-1 the point on X-1 satisfying from a harmonic flag to a double harmonic flag-i.e., a har- a(q-1) = P1 and denote by q1 the point on X0 orthogonal to P1. monic line containing two orthogonal harmonic points. Let yo be the line spanned by q-1 and qj. The vanishing of a We shall describe crossing for n = 6, which generalizes to holomorphic differential implies that the sequence any n. In P5 consider the harmonic sequence a a a A-1 I /0 Pi IPo L-1 - Lo- Li, [4.1] is a Frenet harmonic sequence. This construction is called which is full. By Theorem 3 we have det(ccT) = 0. If cats = 0, turning. By iterating turnings we will come to a Frenet har- 4.1 is a Frenet harmonic sequence and the lines contain monic sequence harmonic points. Consider the remaining case rk(ca,;-3) = 1. Ll is three-di- a a mensional (as a projective space) and L-1, L1 are noninter- secting and nonorthogonal lines in it. The harmonic map S2 G(2, 3) defined by with Pk = kera E v0 but Pk E a(v_1). The line vo can then be shown to be a double harmonic flag containing two orthogo- x-+ Lo(x)', xE S2 nal harmonic points. The inverse procedure to turning, to be called returning, has fundamental collineations that we denote by can be achieved by choosing a number of holomorphic sec- tions of Pl-bundles. 'd, 'a:Lo(x)' -- Lo(x). [4.21 In these constructions some fundamental collineations may vanish identically, in which case we get a holomorphic Our hypothesis says that their restrictions 'aIL 1' IL, are of curve of G(2, n). As a result we may state our theorem as rank 1. We therefore get four geometrically defined points, follows. their images on Lo, and ker('taL l) on L-1, and ker(' lLI) on THEOREM 6. To every harmonic map L:S2 -* G(2, n) there L1. The first two points are orthogonal, and the same is true is associated either (i) a holomorphic curve A:S2 -* G(2, n) of the lines or (ii) a holomorphic curve A:S2 _* P1_ and a harmonic line Ajoining two adjacent vertices ofthe Frenetframe ofA. The Xo = im(aILl) A ker(t alL), map L can be constructedfrom A (in case i) or X (in case ii) by choosing a number of holomorphic sections of Pl-bun- X-1 = im(ILI) A ker('alL_l). [4.3] dles. Further references on our topic can be found in refs. 3 and Let XA be the line, uniquely determined, that is orthogonal to 4. both X0 and X-1. Then we have Theorem 5. THEOREM 5. In the above construction the lines X1 -S Xo S.-s.C. is partially supported by National Science Foundation X1 form a Frenet harmonic sequence. Grants DMS 84-03201 and DMA 84-01959 and J.W. is partially sup- This is a global theorem on S2. Its proof depends on the ported by National Science Foundation Grant DMS 84-05186. differential form of degree 5. The vanishing of a holomorphic 1. Eells, J. & Lemaire, L. (1978) Bull. Lond. Math. Soc. 10, 1-68. association ofthe Frenet harmonic sequence in Theorem 5 to 2. Ramanathan, J. (1984) J. Differ. Geom. 19, 207-219. 4.1 is called crossing. 3. Chern, S. S. & Wolfson, J. G. (1983) Am. J. Math. 105, 59-83. We must understand "recrossing," which is to recover Lo 4. Din, A. M. & Zakrzewski, W. J. (1981) Lett. Math. Phys. 5 from the Frenet harmonic sequence in Theorem 5. It in- (6), 553-561. Downloaded by guest on October 1, 2021