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Segre Embedding and Related Maps and Immersions In Arab J. Math. Sc. Volume 8, Number 1, June 2002 pp. 1-39 SEGRE EMBEDDING AND RELATED MAPS AND IMMERSIONS IN DIFFERENTIAL GEOMETRY BANG-YEN CHEN ABSTRACT. The Segre embedding was introduced by C. Segre (1863-1924) in his famous 1891 article [46]. It plays an important role in algebraic geom­ etry as well as in differential geometry, mathematical physics, and coding theory. In this article, we survey main results on Segre embeddings in dif­ ferential geometry. Moreover, we also present recent differential geometric results on maps and immersions which are constructed in ways similar to Segre embedding. TABLE OF CONTENTS1 1. Introduction. 2. Basic formulas and definitions. 3. Differential geometric characterizations of Segre embedding. 4. Degree of Kahlerian immersions and homogeneous Kahlerian subman- ifolds via Segre embedding. 5. OR-products and Segre embedding. 6. OR-warped products and partial Segre OR-immersions. 7. Real hypersurfaces as partial Segre embeddings. 8. Complex extensors, Lagrangian submanifolds and Segre embedding. 9. Partial Segre OR-immersions in complex projective space. 2000 Mathematics Subject Classification. Primary 53-02, 53C42, 53D12; Secondary 53B25, 53C40, 14E25. Key word and phmses. Segre embedding, totally real submanifold, Lagrangian submani­ fold, CR-submanifold, warped product, CR-warped product, complex projective space, geo­ metric inequality, convolution, Euclidean Segre map, partial Segre map, tensor product immersion, skew Segre embedding. 1The author would like to express his thanks to the editors for the invitation to publish this article in this Journal 2 BANG-YEN CHEN 10. Convolution of Riemannian manifolds. 11. Convolutions and Euclidean Segre maps. 12. Skew Segre embedding. 13. Tensor product immersions and Segre embedding. 14. Conclusion. References 1. INTRODUCTION Throughout this article, we denote by cpn(c) the complex projective n­ space endowed with the Fubini-Study metric with constant holomorphic sec­ tional curvature c. Let (zo, ... , Zn) denote a homogeneous coordinate system on cpn(c) There are two well-known examples of algebraic manifolds in complex pro­ jective spaces: the Veronese embedding Vn: cpn(2)-+ Qpn(n+3)12 (4) and the Segre embedding Shp: CPh(4) X C?P(4)-+ cph+p+hP(4). The Veronese embedding Vn is a Kahlerian embedding, that is a holomor­ phically isometric embedding, of cpn(2) into cpn(n+3)/2 (4) given by homo­ geneous monomials of degree 2: with O:i + O:j = 2. For n = 1, this is nothing but the quadric curve Ql = {(zo, ZI, Z2) E CP2 : tzJ = 0} J=O The Veronese embedding can be extended to o:-th Veronese embedding v~ with o: ~ 2: (1.2) SEGRE EMBEDDING IN DIFFERENTIAL GEOMETRY 3 defined by 1 (1.3) (zo, ... ,zn) f-4 (z0,y'az0- zl, ... , ./ 1 o: 1zgo ... z~n, ... ,z~) vO:Q •••• O:n. with o:o + · · · + O:n = o:. On the other hand, the Segre embedding (1.4) Shp: CPh(4) x CPP(4)---* cph+P+hP(4), is defined by = o<t< , (1.5) Shp(ZO, ... , zh; wo, ... , wp) (ziwt)o<_]_ "<h ' - _p where (zo, ... , zh) and (wo, ... , wp) are the homogeneous coordinates of CPh(4) and C?P(4), respectively. This embedding (1.4) was introduced by C. Segre in 1891 (see [46]). It is well-known that the Segre embedding Shp is also a Kahlerian embedding. When h = p = 1, the Segre embedding is nothing but the complex quadric surface, Q2 = C P 1 x C P 1 in C P 3 , defined by (1.6) Q2 = {(zo, z1, z2, z3) E cp3: tzJ = o}. ]=0 The Segre embedding can also be naturally extended to productembeddings of arbitrary number of complex projective spaces as follows. Let ( zb, ... , z~i) ( 1 :=:; i :=:; s) de~ote the homogeneous coordinates of C pni. Define a map: (1. 7) s N= IJ(ni+1)-1, i=l which maps each point ((z6, ... , z~J, ... , (z0, ... , z~J) in the product Kah- lerian manifold CPn1 ( 4) X··· xCpn• (4) to the point (z{1 · · · zf)lsiisn1 , ... ,1si.sn. in CPN(4). This map Sn 1 ... n. is also a Kahlerian embedding. The Segre embedding is known to be the simplest Kahlerian embedding from product algebraic manifolds into complex projective spaces. It is well-known that the Segre embedding plays an important role in algebraic geometry (see 4 BANG-YEN CHEN [37, 49]). The Segre embedding has also been applied to differential geometry as well as to coding theory (see, for instance, [32, 45, 48]) and to mathematical physics (see, for instance, [4, 4 7]). The purpose of this article is to survey the main results on Segre embed­ ding in differential geometry. Furthermore, we also present recent results in differential geometry concerning maps, immersions, and embedding which are constructed in ways similar to Segre embedding defined by (1.5). 2. BASIC FORMULAS AND DEFINITIONS Let M be a Riemannian n-manifold with inner product(,) and let e1, ... , en be an orthonormal frame field on M. For a differentiable function <p on M, the gradient V' <p and the Laplacian l:ir.p of <p are defined respectively by (2.1) (Y' r.p, X) = X r.p, n (2.2) l:ir.p = L {ejej<p- (Y' eiei )r.p} j=l for vector fields X tangent to M, where V' is the Levi-Civita connection on M. If M is isometrically immersed in a Riemannian manifold M, then the formulas of Gauss and Weingarten forM in Mare given respectively by (2.3) VxY = 'VxY + a(X, Y), (2.4) '\7 x~ = -A~ X+ Dx~ for vector fields X, Y tangent toM and~ normal toM, where '\7 denotes the Levi-Civita connection on M, a the second fundamental form, D the normal connection, and A the shape operator of Min M. The second fundamental form and the shape operator are related by ~ The mean curvature vector. H is defined by ~ 1 1 n (2.5) H =-trace a=-~ h(ei, ei), n n~ i=l SEGRE EMBEDDING IN DIFFERENTIAL GEOMETRY 5 where { e1, ... , en} is a local orthonormal frame of the tangent bundle T M of M. The squared mean curvature is given by H 2 = (H., H.). A submanifold M is called totally geodesic in M if the second fundamental form of M in M vanishes identically. And M is called minimal if its mean curvature vector vanishes identically. The equation of Gauss is given by (2.6) R(X, Y; Z, W) = R(X, Y; Z, W)+ < u(X, Z), u(Y, W) - (u(X, W), u(Y, Z)) , for X, Y, Z, W tangent toM, where Rand R denote the curvature tensors of M and M, respectively. For the second fundamental form u, we define its covariant derivative "Vu with respect to the connection on T M EB Tl_ M by (2.7) (\/' xu)(Y, Z) = Dx(u(Y, Z)) - u('\1 x Y, Z) - u(Y, '\1 x Z) for X, Y, Z tangent toM. The equation of Codazzi is (2.8) (R(X, Y)Z)j_ = (\/' xu)(Y, Z) - (\/'yu)(X, Z), where (R(X, Y)Z)l_ denotes the normal component of R(X, Y)Z. A submanifold M in a Riemannian manifold M is said to have parallel second fundamental form if Vu = 0 identically. The Riemann curvature tensor of a complex space form _M-m(4c) of constant holomorphic sectional curvature 4c is given by (2.9) R(X, Y; Z, W) = c {(X, W) (Y, Z) -(X, Z) (Y, W) + (JX, W) (JY, Z)- (JX, Z) (JY, W) + 2 (X, JY) (JZ, W) }. If we define the k-th (k 2: 1) covariant derivative of the second fundamental form u by 1 (2.10) (\l'ku)(Xt, ... , xk+2) = Dxk+2 ((\/'k- u)(X1, ... , xk+l)) k+l - I:<(vk-1u)(Xl, ... , "Vxk+ 2 Xi, ... , xk+I), i=l 6 BANG-YEN CHEN -k then \7 u is a normal-bundle-valued tensor of type (0, k + 2). Moreover, it can -k be proved that \7 u satisfies (2.11) (Vku)(X1,X2,X3, ... ,Xk+2)- (Vku)(X2,X1,X3, ... ,Xk+2) = R D (X1,X2) ((-\7 k-2 u ) (X3, ... ,Xk+2) ) k+2 + L) (Vk- 2 u) (X3, ... , R(X1, X2)Xi, ... , Xk+2), i=3 fork~ 2. For simplicity, we put V0u = u. 3. DIFFERENTIAL GEOMETRIC CHARACTERIZATIONS OF THE SEGRE EMBEDDING H. Nakagawa and R. Tagaki [38] classify complete Kahlerian submanifold in complex projective spaces with parallel second fundamental form. In particu­ lar, they show that the Segre embedding are the only one which are reducible. More precisely, they prove the following. Theorem 3.1. Let M be a complete Kiihlerian submanifold rf complex dimen­ sion n embedded in CPm(4). If M is reducible and has parallel second funda­ mental form, then M is congruent to Cpn1 (4) x Qpn2 (4) with n = n1 + n2, and the embedding is given by the Segre embedding. The following results, published in 1981 by B. Y. Chen [6] for s = 2 and by B. Y. Chen and W. E. Kuan [23, 24] for s ~ 3, can be regarded as a "converse" to Segre embedding constructed in 1891 by C. Segre. Theorem 3.2 ([6, 23, 24]). Let Mf1 , ••• , M:• be Kiihlerian manifolds of di­ mensions n1, ... , n 8 , respectively. Then, locally, every Kiihlerian immersion s f : Mf1 x .. · x M:· ---> C pN ( 4), N = IJ (ni + 1) - 1, i=l 1 of Mf x · · · x M:• into C pN ( 4) is a Segre embedding, that is, Mf1 , ••• , M:• are open portions of CPn1 (4), ... , CPn•(4), respectively. Moreover, the Kiih­ lerian immersion f is given by a Segre embedding. SEGRE EMBEDDING IN DIFFERENTIAL GEOMETRY 7 Let IIVk all2 denote the squared norm of the k-th covariant derivative of the second fundamental form.
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