Proc. Natl. Acad. Sci. USA Vol. 82, pp. 2217-2219, April 1985 Mathematics Harmonic maps of S2 into a complex Grassmann manifold (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 spaces of dimensions k and n, respec- tary group 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] space Pn-1 itself. We study harmonic maps of the two-dimen- sional sphere S2 into G(k, n). The case k = 1 has been the Taking the exterior derivative of Eq. 1.6, we get the Maurer- subject of investigation by several authors [see, for example, Cartan equations 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 geometry 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 multivector 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 map to new harmonic maps, in which called such a structure a Segre structure.) In particular, the the image 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 metric. 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 connection forms are w,,j,coo; their exterior derivatives 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 surface 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 absolute value 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] point 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 dimension, 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 equation 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 exterior derivative 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 curve 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 differential form. 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 projective space of dimension k - THEOREM4.