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publish publish to to
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author author The The 1
embedding. embedding. Segre Segre skew skew immersion, immersion,
product product
tensor tensor map, map,
Segre Segre
partial partial map, map, Segre Segre Euclidean Euclidean convolution, convolution, inequality, inequality, metric metric
geo
space, space, projective projective
complex complex product, product, CR-warped CR-warped product, product, warped warped CR-submanifold, CR-submanifold, fold, fold,
submani
Lagrangian Lagrangian
submanifold, submanifold, real real totally totally embedding, embedding, Segre Segre phmses. phmses. and and word word Key Key
14E25. 14E25. 53C40, 53C40, 53B25, 53B25,
Secondary Secondary
53D12; 53D12;
53C42, 53C42, 53-02, 53-02, Primary Primary Classification. Classification. Subject Subject Mathematics Mathematics 2000 2000
space. space. projective projective complex complex in in OR-immersions OR-immersions Segre Segre Partial Partial 9. 9.
embedding. embedding. Segre Segre and and submanifolds submanifolds Lagrangian Lagrangian extensors, extensors, Complex Complex 8. 8.
embeddings. embeddings. Segre Segre partial partial as as hypersurfaces hypersurfaces Real Real 7. 7.
OR-immersions. OR-immersions. Segre Segre partial partial and and products products OR-warped OR-warped 6. 6.
embedding. embedding. Segre Segre and and OR-products OR-products 5. 5.
embedding. embedding. Segre Segre via via ifolds ifolds
subman-
Kahlerian Kahlerian homogeneous homogeneous and and immersions immersions Kahlerian Kahlerian of of Degree Degree 4. 4.
embedding. embedding. Segre Segre of of characterizations characterizations geometric geometric Differential Differential 3. 3.
definitions. definitions. and and formulas formulas Basic Basic 2. 2.
Introduction. Introduction. 1. 1.
CONTENTS OF OF TABLE TABLE 1 1
embedding. embedding. Segre Segre
to to
similar similar
ways ways in in constructed constructed are are which which immersions immersions and and maps maps on on results results
geometric geometric
differential differential recent recent present present also also we we Moreover, Moreover, geometry. geometry. ferential ferential
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physics, physics, mathematical mathematical geometry, geometry, differential differential in in as as well well as as etry etry
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algebraic algebraic in in role role important important an an plays plays It It [46]. [46]. article article 1891 1891 famous famous his his in in
(1863-1924) (1863-1924) Segre Segre C. C. by by introduced introduced was was embedding embedding Segre Segre The The ABSTRACT. ABSTRACT.
CHEN CHEN BANG-YEN BANG-YEN
GEOMETRY GEOMETRY DIFFERENTIAL DIFFERENTIAL IN IN IMMERSIONS IMMERSIONS
AND AND MAPS MAPS
RELATED RELATED AND AND EMBEDDING EMBEDDING SEGRE SEGRE
1-39 1-39 pp. pp.
2002 2002 June June 1, 1, Number Number 8, 8, Volume Volume
Sc. Sc. Math. Math. J. J. Arab Arab 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)
that that the the Segre Segre embedding embedding plays plays an an important important role role in in algebraic algebraic geometry geometry (see (see
product product algebraic algebraic manifolds manifolds into into complex complex projective projective spaces. spaces. It It is is well-known well-known
The The Segre Segre embedding embedding is is known known to to be be the the simplest simplest Kahlerian Kahlerian embedding embedding from from
in in 1 1 CPN(4). CPN(4). This This Sn n. n. map map is is also also a a Kahlerian Kahlerian embedding. embedding......
lerian lerian
CPn 1 1 manifold manifold
xCpn• xCpn• X··· X··· 4) 4) (4) (4) to to the the point point (z{ zf)lsiisn 1 ,1si.sn. ,1si.sn. ( ( · · · · · · , , ......
1 1
0 ...... , , which which maps maps , , ...... each each , , point point ((z6, ((z6, ...... , , z~J, z~J, (z z~J) z~J) in in the the product product Kah-
i=l i=l
IJ(ni+1)-1, IJ(ni+1)-1, N= N=
s s
(1. (1. 7) 7)
Define Define a a map: map:
zb, zb, ...... , , ( ( Let Let ( ( s) s) z~i) z~i) 1 1 :=:; :=:; :=:; :=:; de~ote de~ote i i the the homogeneous homogeneous coordinates coordinates pni. pni. C C of of
of of arbitrary arbitrary number number of of complex complex projective projective spaces spaces as as follows. follows.
The The Segre Segre embedding embedding can can also also be be naturally naturally extended extended to to productembeddings productembeddings
]=0 ]=0
tzJ tzJ
cp3: cp3:
o}. o}. = = {(zo, {(zo, (1.6) (1.6) Q2 Q2 = = z1, z2, z2, z1, z3) z3) E E
= = Q surface, surface, 2 2 C C C C
C C P x x
P in in P defined defined by by , , 1 1 1 1 3
= = = = When When h h 1, 1, the the Segre Segre p p embedding embedding is is nothing nothing but but the the complex complex quadric quadric
Kahlerian Kahlerian embedding. embedding.
1891 1891 in in [46]). [46]). (see (see It It is is well-known well-known that that the the Segre Segre embedding embedding Shp Shp is is also also a a
C?P(4), C?P(4), and and respectively. respectively. This This embedding embedding (1.4) (1.4) was was introduced introduced by by C. C. Segre Segre
(zo, (zo, ...... , , where where zh) zh) (wo, (wo, ...... and and , , wp) wp) are are the the homogeneous homogeneous coordinates coordinates CPh(4) CPh(4) of of
_]_ _]_ - ' _p _p
(1.5) (1.5) Shp(ZO, Shp(ZO, ...... , , zh; zh; wo, wo, ...... , , wp) wp) (ziwt)o< (ziwt)o< " is is defined defined by by (1.4) (1.4) Shp: Shp: CPh(4) CPh(4) CPP(4)---* CPP(4)---* cph+P+hP(4), cph+P+hP(4), x x On On the the other other hand, hand, the the Segre Segre embedding embedding + + · · · · · · + + with with o:o o:o o:. o:. = = O:n O:n v v O:Q O:Q •••• •••• O:n. O:n. 0 0 z~n, z~n, ,z~) ,z~) 1 1 1 - (1.3) (1.3) (zo, (zo, ...... ,zn) ,zn) (z ,y'az zl, zl, ...... , , ./ ./ zgo zgo ...... o: o: ...... f-4 f-4 1 defined defined by by SEGRE SEGRE EMBEDDING EMBEDDING IN IN DIFFERENTIAL DIFFERENTIAL GEOMETRY GEOMETRY 3 3 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 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 i=l i=l - I:<(vk- u)(Xl, u)(Xl, 2 ...... xk+I), xk+I), , , "Vxk+ Xi, Xi, ...... , , 1 k+l k+l (2.10) (2.10) (\l'ku)(Xt, (\l'ku)(Xt, xk+2) xk+2) 2 2 ...... , , ((\/'k- Dxk+ = = u)(X1, u)(X1, ...... , , xk+l)) xk+l)) 1 form form u u by by we we If If define define 2: 2: the the (k (k k-th k-th 1) 1) covariant covariant derivative derivative of of the the second second fundamental fundamental 2 2 (X, (X, JY) JY) W) W) (JZ, (JZ, }. }. W) W) (JX, (JX, + + Z)- (JY, (JY, Z) Z) (JX, (JX, W) W) (JY, (JY, (2.9) (2.9) R(X, R(X, W) W) Y; Y; {(X, {(X, Z, Z, c c W) W) (Y, (Y, + + Z) Z) = = -(X, -(X, Z) Z) (Y, (Y, W) W) holomorphic holomorphic sectional sectional curvature curvature 4c 4c is is given given by by The The Riemann Riemann curvature curvature tensor tensor of of a a complex complex space space _M-m(4c) _M-m(4c) form form of of constant constant second second fundamental fundamental Vu Vu form form if if 0 0 identically. identically. = = A A submanifold submanifold M M M M in in a a Riemannian Riemannian manifold manifold is is said said to to have have parallel parallel where where (R(X, (R(X, Y)Z)l_ Y)Z)l_ denotes denotes the the normal normal component component of of R(X, R(X, Y)Z. Y)Z. (2.8) (2.8) (R(X, (R(X, Y)Z)j_ Y)Z)j_ xu)(Y, xu)(Y, Z) Z) - (\/'yu)(X, (\/'yu)(X, = = (\/' (\/' Z), Z), for for X, X, Y, Y, Z Z tangent tangent toM. toM. The The equation equation of of Codazzi Codazzi is is (2.7) (2.7) (\/' (\/' Z) Z) xu)(Y, xu)(Y, Dx(u(Y, Dx(u(Y, Z)) Z)) = = - Z) Z) u('\1 u('\1 - x x Y, Y, Z) Z) u(Y, u(Y, '\1 '\1 x x with with respect respect to to the the connection connection on on EB EB T M M T Tl_ Tl_ by by M M For For the the second second fundamental fundamental form form we we u, u, define define its its covariant covariant derivative derivative "Vu "Vu M, M, M M and and respectively. respectively. for for R R X, X, tangent tangent Y, Y, Z, Z, W W toM, toM, where where Rand Rand denote denote the the curvature curvature tensors tensors of of - (u(X, (u(X, W), W), Z)) Z)) , , u(Y, u(Y, (2.6) (2.6) R(X, R(X, Y; Y; Z, Z, W) W) < < R(X, R(X, = = Y; Y; Z, Z, W)+ W)+ u(X, u(X, Z), Z), u(Y, u(Y, W) W) The The equation equation of of Gauss Gauss is is given given by by vanishes vanishes identically. identically. vanishes vanishes identically. identically. And And M M is is called called minimal minimal if if its its mean mean curvature curvature vector vector M M is is called called totally totally M M geodesic geodesic in in if if the the second second fundamental fundamental M M form form of of M M in in (H., (H., H.). H.). M. M. The The squared squared mean mean curvature curvature is is given given by by H = = A A submanifold submanifold 2 2 where where en} en} { { e1, e1, ...... , , is is a a local local orthonormal orthonormal frame frame of of the the tangent tangent bundle bundle T M M T of of SEGRE SEGRE EMBEDDING EMBEDDING IN IN DIFFERENTIAL DIFFERENTIAL GEOMETRY GEOMETRY 5 5 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. = = = = ...... w w cn-s+I, cn-s+I, E E ,Wn-s-1) ,Wn-s-1) (w~, (w~, C ,z E E ...... (z~, (z~, z z where where [(z,w)], [(z,w)], by by 8 , , ) ) 8 represented represented be be can can CPJl'(c) CPJl'(c) of of point point a a Thus Thus 1. 1. curvature curvature sectional sectional constant constant of of and and 2s 2s index index with with unit-sphere unit-sphere indefinite indefinite !)-dimensional !)-dimensional 8~~+1(1) 8~~+1(1) (2n+ (2n+ the the be be let let and and 4, 4, curvature curvature sectional sectional holomorphic holomorphic constant constant and and 2s, 2s, index index n, n, mension mension di complex complex of of space space projective projective complex complex indefinite indefinite the the denote denote Let Let 0~(4) 0~(4) Sh,p· Sh,p· embedding embedding Segre Segre the the by by given given is is immersion immersion Kiihlerian Kiihlerian the the and and CFP(4), CFP(4), respectively, respectively, and and ofCPh(4) ofCPh(4) Mf Mf Mf Mf portions portions open open are are and and if if only only and and if if (3.2) (3.2) holds holds of of sign sign equality equality The The 2:: 2:: llaW llaW 8hp. 8hp. (3.2) (3.2) have have we we Then Then codimension. codimension. arbitrary arbitrary of of cpm(4) cpm(4) Mf Mf in in submanifold submanifold Mf Mf Kiihlerian Kiihlerian ([6]). ([6]). product product a a be be Let Let x x 3.5 3.5 Theorem Theorem embedding. embedding. Segre Segre the the by by 'given 'given is is f f immersion immersion Kiihlerian Kiihlerian Mf Mf the the moreover, moreover, and and respectively, respectively, CFP, CFP, and and Qph Qph of of portions portions open open are are and and Mf Mf Mf Mf Mf Mf is, is, that that embedding, embedding, Segre Segre the the locally locally is is Qph+p+hP(4) Qph+p+hP(4) into into of of x x Mf---+ Mf---+ f: f: Mf Mf cph+p+hP(4), cph+p+hP(4), x x immersion immersion Kiihlerian Kiihlerian every every Then Then respectively. respectively. p, p, and and h h dimensions dimensions Mf Mf complex complex of of Mf Mf manifolds manifolds ([6]). ([6]). Kiihlerian Kiihlerian two two be be and and Let Let 3.4 3.4 Theorem Theorem = = 3.3 3.3 to to reduce reduce 3.2 3.2 Theorem Theorem and and Theorem Theorem 2, 2, When When k k embedding. embedding. Segre Segre the the by by given given is is immersion immersion Kiihlerian Kiihlerian the the and and respectively, respectively, ,CP"'•(4), ,CP"'•(4), ...... (4), (4), ofCPn portions portions open open are are 1 Mf M-;• M-;• 2:: 2:: if if only only and and if if 2 2 k k some some for for (3.1) (3.1) holds holds of of sign sign equality equality The The , , ••• ••• , , 1 ·. ·. · · · · 2,3, 2,3, fork= fork= L L 2:: 2:: 2k 2k aW aW 11Vk- k! k! nk, nk, (3.1) (3.1) · · · · · · n1 n1 2 have have we we Then Then codimension. codimension. arbitrary arbitrary of of CPm(4) CPm(4) in in submanifold submanifold Mf Kiihlerian Kiihlerian product product a a x x be be · · · · ([6, ([6, · · x x M-;• M-;• 24]). 24]). Let Let 23, 23, 3.3 3.3 Theorem Theorem 1 1 following. following. the the in in 11Vkall given given as as 2 2 by by characterized characterized be be also also can can embedding embedding Segre Segre The The form. form. fundamental fundamental second second the the of of derivative derivative covariant covariant k-th k-th the the of of norm norm squared squared all the the IIVk IIVk denote denote Let Let 2 2 7 7 GEOMETRY GEOMETRY DIFFERENTIAL DIFFERENTIAL IN IN EMBEDDING EMBEDDING SEGRE SEGRE 8 BANG-YEN CHEN (z, w) E 5~~+ 1 (1) C c~+l and [(z, w)] is the equivalent class of the Hopf projection Consider the map ¢: CP~(4) x CPf(4)---> C~~:.~,t)(4) with N(h,p)=h+p+hp, R(h,p, s,t) = s(p- t) + t(h- s) + s + t, given by for 1 :S i,j :S s; 1 :S k, C :S h- s + 1; 1 :Sa, b :S t; 1 :Sa, j3 :S h- t + 1. Then ¢ is a well-defined holomorphic isometric embedding, which is called the indefinite Segre embedding CP:(4) x CPf(4) into C~(~,p),p,s, t)(4). T. Ikawa, H. Nakagawa and A. Romero [31] studied the indefinite version of Theorem 3.4 and obtained the following result. Theorem 3.6. Let M:; and Mf' be two complete indefinite Kiihlerian man ifolds with complex dimensions n and m, and indices 2s and 2t, respectively. If there exists a holomorphic isometric immersion from the product M:; X Mf' into an indefinite complex projective space of complex dimension N and index 2r, then we have: (1) N 2: n + m + nm and r 2: s(m- t) + t(n- s) + s + t. (2) If N = n + m + nm, then the immersion is obtained by the indefinite Segre imbedding. Remark 3.1. The assumption of "Kahlerian immersion" in Theorems 3.2, 3.3, and 3.4 is necessary. In fact, let Mi be a projective nonsingular embedded variety of dimension ni 2: 1 (i = 1, 2, · · · , r) and let M = Ml X 0 0 0 X Mr c cpn! X 0 0 0 X cpnr __sn_l_··_·n_r_4 cpN Segre embedding = = rank(Mi)· rank(Mi)· ri ri where where I:::= I:::= , , ...... !I, !I, · · ripi, ripi, by by given given Is Is of of ·~Is ·~Is !I~· !I~· product product is is tensor tensor the the of of degree degree the the I I Then Then type. type. compact compact of of spaces spaces symmetric symmetric Hermitian Hermitian irreducible irreducible of of embeddings embeddings Kiihlerian Kiihlerian full full Pi-th Pi-th be be s, s, , , ...... 1, 1, Mi Mi : : i i fi fi Let Let CPNi(4), CPNi(4), 4.1. 4.1. = = Theorem Theorem --+- result. result. following following the the proved proved they they namely, namely, space; space; projective projective complex complex in in submanifold submanifold Kahlerian Kahlerian symmetric symmetric a a of of rank rank the the and and degree degree the the between between relation close close a a obtained obtained had had [50] [50] in in Takeuchi Takeuchi M. M. and and R. R. Tagaki Tagaki and and [38] [38] in in Tagaki Tagaki R. R. and and Nakagawa Nakagawa H. H. , , ...... JI, JI, fs· fs· of of product product tensor tensor the the called called is is which which f1:= + + 1 embedding, embedding, Kahlerian Kahlerian full full a a is is composition composition This This (Ni (Ni 1) 1) -1. -1. with with N N = = ' ' embedding embedding Segre Segre 4 ( cpN cpN ) ) ·Ns ·Ns .. .. SNt SNt embedding embedding product product (4.1) (4.1) xCP. xCP. Ms Ms :Ml :Ml -----~CP -----~CP ...... X X h~···~!s h~···~!s X X ...... X X 1 1 N N fs fs N N X X X··· X··· /1 /1 by by given given composition the the Consider Consider type. type. compact compact of of spaces spaces symmetric symmetric Hermitian Hermitian irreducible irreducible of of dings dings embed Kahlerian Kahlerian full full are are ,s, ,s, ...... 1, 1, Mi--+- CPN•(4), CPN•(4), i i fi: fi: that that Suppose Suppose = = follows: follows: as as spaces spaces projective projective complex complex in in immersions immersions Kahlerian Kahlerian of of product product tensor tensor a a of of notion notion the the [50] [50] in in define define Takeuchi Takeuchi M. M. and and R. R. Takagi Takagi embeddings, embeddings, Segre Segre Using Using EMBEDDING EMBEDDING SEGRE SEGRE VIA VIA SUBMANIFOLDS SUBMANIFOLDS KAHLERIAN KAHLERIAN HOMOGENEOUS HOMOGENEOUS AND AND IMMERSIONS IMMERSIONS KAHLERIAN KAHLERIAN OF OF DEGREE DEGREE 4. 4. + + = = - n2) n2) 2(nl 2(nl e e 1. 1. case case ======+ + ...... + + = = which which in in x2 x2 xl xl 2, 2, CPn +nr) +nr) unless unless cpnt, cpnt, 1, 1, e e 2(nl 2(nl that that r r , , 2 M. M. proves proves He He of of classes classes Segre Segre the the of of degree degree the the of of terms terms in in e e characterizes characterizes Holm~, Holm~, Dale Dale cpe-I, cpe-I, A. A. of of result result algebraic algebraic an an Using Using projection. projection. a a via via in in cpe, cpe, not not but but in in general) general) in in embedded embedded Kahlerian Kahlerian necessary necessary (not (not embedded embedded be be can can that that such such M M e e dimension dimension embedding embedding the the finding finding of of problem problem the the [27] [27] ( ( + + in in considers considers Dale Dale M. M. embedding. embedding. Segre Segre the the via via - 1 1 - 1) 1) TI~=l TI~=l ni ni N N with with = = Mr Mr into into cpN cpN product product Ml Ml the the from from embedding embedding composition composition the the be be X X ...... X X 9 9 GEOMETRY GEOMETRY DIFFERENTIAL DIFFERENTIAL IN IN EMBEDDING EMBEDDING SEGRE SEGRE 10 BANG-YEN CHEN Related to this theorem, we mention the following result by M. Takeuchi [51] for Kahlerian immersions of homogeneous Kahlerian manifolds. Theorem 4.2. Let f: M ~ CPm(4) be a Kiihlerian immersion of a globally homogeneous K iihlerian manifold M. Then (1) M is compact and simply-connected; (2) f is an embedding; and (3) M is the orbit in cpm(4) of the highest weight in an irreducible unitary representation of a compact semisimple Lie group. The notion of the degree of Kahlerian immersions in the sense of [50] is defined as follows: Let V be a real vector space of dimension 2n with an almost complex structure J and a Hermitian inner product g. Denote the complex linear extensions of J and g to the complexification V 0 of V by the same J and g, respectively. Let v+ and v- be the eigensubspace of Jon V 0 with eigenvalue 1 and -1, respectively. Then V 0 = v+ EB v- is an orthogonal direct sum with respect to the inner product. Let E be a real vector bundle over a manifold M with a Hermitian structure ( J, g) on fibres. The Hermitian structure induces a Hermitian inner product on E 0 . We have subbundles E+ and E- such that E 0 satisfies E 0 = E+ EBE and the complex conjugation E± ~ E"f. The map on the space of sections induced from the complex conjugation is denoted by f(E±) ~ f(E'f'). Let (M, g, J) be a Kahlerian manifold. Then we get a Hermitian inner product on the complexification T(M)0 and subbundles T(M)± of T(M)0 such that T(M)0 = T(M)+ EB T(M)-. Let f : (M, g, J) ~ (M', g', J') be a Kahlerian immersion between Kahlerian manifolds. The Levi-Civita connections of M and M' are denoted by V' and V''. The induced bundle J*T(M') has a Hermitian structure ( J', g') induced from the one on M'. It also has a connection V'' induced from M'. If we denote the orthogonal complement of f.T(x(M) in Tf(x)(M') by Tj:(M), then Tj_(M) is a subbundle of f*T(M'). We have the orthogonal Whitney sum decompositions: f*T(M') = f.T(M) EB Tj_(M), J*T(M')0 = f.T(M) 0 EB Tj_(M) 0 , and f*T(M')± = f.T(M)± EB Tj_(M)±. we we have have an an orthogonal orthogonal direct direct sum: sum: ment ment 'H.~- of of (M) (M) 1-l~(M), 1-l~(M), in in where where 1-l~(M) 1-l~(M) is is understood understood to to be be {0}. {0}. Then Then 1 of of increasing increasing Tf(x)(M')+. Tf(x)(M')+. subspaces subspaces of of O~(M) O~(M) Let Let be be the the orthogonal orthogonal comple c c c c 1-l~ 1-l~ · · · · · · c c c c 1-l~ 1-l~ 1-f.~+l 1-f.~+l c c · · · · · · 1-l! 1-l! c c (M) (M) (M) (M) (M) (M) Tt(x) Tt(x) (M) (M) (M')+ (M')+ series series 9 9 2 2: H( H( subspace subspace Tx(M)+ Tx(M)+ spanned spanned by by ®iTx(M)+). ®iTx(M)+). and and Then Then we we get get a a :-:;j > > k k For For an an integer integer 0, 0, we we define define a a subspace subspace Tt(x)(M')+ Tt(x)(M')+ 1-l~(M) 1-l~(M) of of to to be be the the k~2 k~2 k~2 k~2 H= H= LHk LHk r(Hom(L®kT(M)+,Tl_(M)+))· r(Hom(L®kT(M)+,Tl_(M)+))· E E k~2 k~2 k~2 k~2 h h L(J'k L(J'k r(Hom(L:::®kT(M),Tj_(M))), r(Hom(L:::®kT(M),Tj_(M))), = = E E We We put put Hk(Xt, Hk(Xt, ...... , , Xk) Xk) uk(Xt, uk(Xt, Tx(M)+. Tx(M)+...... , , Xk), Xk), Xi Xi = = E E Let Let Hk Hk r(Hom(®kT(M)+,Tl_(M)+)) r(Hom(®kT(M)+,Tl_(M)+)) 2) 2) (k (k E E be be ~ ~ defined defined by by Tx(M)+ Tx(M)+ Tx(M)c. Tx(M)c. and and Xa, Xa, ...... , , Xk Xk E E (4.2) (4.2) (4.3) (4.3) Equations Equations and and imply imply that that T}(M)+ T}(M)+ uk(Xt, uk(Xt, ...... ,Xk) ,Xk) for for X1,X2 X1,X2 E E E E Tx(M). Tx(M). for for Xi Xi E E i=2 i=2 xi, xi, ...... ' ' - L L uk(Xt, uk(Xt, ...... 'V' 'V' Xk), Xk), xk+l xk+l k k = = uk+I(Xt, uk+I(Xt, ...... ,Xk+t) ,Xk+t) 1 uk(Xt, uk(Xt, (4.3) (4.3) ...... ,Xk) ,Xk) Dxk+ Put Put u2 u2 u. u. 3, 3, we we define define (2.10) (2.10) uk uk inductively inductively Fork~ Fork~ = = just just like like by by Let Let denote denote f. f. u u the the second second fundamental fundamental form form of of We We have have normal normal Tl_(M) Tl_(M) connection connection Don Don satisfies satisfies (v:Xe)j_. (v:Xe)j_. Dxe Dxe = = the the induced induced projection projection f(f*T(M')) f(f*T(M')) ~ ~ f(Tj_(M)) f(Tj_(M)) is is el_. el_. denoted denoted bye~ bye~ The The The The orthogonal orthogonal f*T(M') f*T(M') projection projection Tl_(M) Tl_(M) ~ ~ is is denoted denoted X~ X~ by by Xj_ Xj_ and and SEGRE SEGRE EMBEDDING EMBEDDING IN IN DIFFERENTIAL DIFFERENTIAL GEOMETRY GEOMETRY 11 11 12 BANG-YEN CHEN Define R1 = M. For an integer k > 1, we define the set Rk of k-regular points of M inductively by Rk = {u E Rk-I: dime 1-l!(M) = max dime 1-l;(M)}. yE'Rk-1 Then we have the inclusions: R1 :J R2 :J · · · :J Rk :J 'Rk+l :J · · · . Note that each Rk is an open nonempty subset of M and, for each k, 1-lk(M) = UxE'R.K 1-l~(M) is a complex vector bundle over Rk, which is a subbundle of f*T(M')+iRk. For an integer k 2: 1 and a point x E Rk, we have [50] (1) V'~Y E 1-l~+l(M) for X E Tx(M)+ and local sections Y of 1-lk(M); (2) O~+l(M) = {0} if and only if, for each X E Tx(M)+ and each local section Y of 1-lk(M), we have V'~Y E 1-l~(M). Thus, there is a unique integer d > 0 such that O~(M) =/= {0} for some X E Rd and o~+I(M) = {0} for each X E Rd. The integer dis called the degree of the Kahlerian immersion f : M --+ M. 5. OR-PRODUCTS AND SEGRE EMBEDDING A submanifold N in a Kahlerian manifold M is called a totally real sub manifold [26] if the complex structure J of M carries each tangent space of N into its corresponding normal space, that is, JTxN C Tf N, x E N. An n-dimensional totally real submanifold in a Kahlerian manifold ifn with com plex dimension n is called a Lagrangian submanifold. (For the latest surveys on Lagrangian submanifolds from a differential geometric point of view, see [13, 15]). A submanifold N in a Kahlerian manifold M is called a 0 R-submanifold [2] if there exists on N a holomorphic distribution 1J whose orthogonal com plement 'D_l_ is a totally real distribution, that is, J'D; C Tf N. The notion of OR-products was introduced in [6] as follows: A OR-submani fold N of a Kahlerian manifold M is called a OR-product if it is locally a subspaces. subspaces. complex complex linear linear of of submanifold submanifold real real totally totally a a and and submanifold submanifold Kiihlerian Kiihlerian a a of of sum sum direct direct a a is is it it if if only only and and if if OR-product OR-product a a is is em em m-space m-space Euclidean Euclidean ([6]). ([6]). complex complex the the in in OR-submanifold OR-submanifold A A 5.1 5.1 Theorem Theorem known. known. are are results results following following the the forms, forms, space space complex complex in in OR-products OR-products For For [6]. [6]. OR-products OR-products a a standard standard called called is is which which oph+p+hP(4) oph+p+hP(4) in in OR-product OR-product a a is is embedding embedding Segre Segre oph+PhP(4) oph+PhP(4) Shp Shp immersion immersion product product (5.2) (5.2) oph(4) oph(4) OPh(4) OPh(4) x x __ __ N1_ N1_ (t,r.p): (t,r.p): OPh(4) OPh(4) x x shp shp o o <_L,r.p_)_--+ <_L,r.p_)_--+ composition composition the the then then O.PP(4), O.PP(4), into into N1_ N1_ p-manifold p-manifold Riemannian Riemannian a a of of immersion immersion Lagrangian Lagrangian a a is is O.PP(4) O.PP(4) r.p: r.p: N1_ N1_ ~ ~ and and 0Ph(4) 0Ph(4) of of map map identity identity the the is is ~ ~ 0Ph(4) 0Ph(4) 0Ph(4) 0Ph(4) if if particular, particular, t: t: In In opnt+n2+n1n2(4). opnt+n2+n1n2(4). in in OR-product OR-product a a as as immersed immersed isometric isometric is is embedding embedding Segre Segre opnt+n2+n1n2(4) opnt+n2+n1n2(4) Sn1n2 Sn1n2 immersion immersion product product (5.1) (5.1) X X (4) (4) X X 0Pn (4) (4) 'I/J2): 'I/J2): opn ('1/Jl, ('1/Jl, _) _) __ __ Nl_ Nl_ Nr Nr {1/J_l_,1f;_ ---+ ---+ 0 0 1 2 Sntn2 Sntn2 2 tion tion composi the the Then Then (4). (4). 0Pn into into N1_ N1_ p-manifold p-manifold Riemannian Riemannian a a of of immersion immersion 2 2 real real totally totally a a be be (4) (4) opn N1_ N1_ let let and and ~ ~ (4) (4) opn into into 'lj; Nr Nr manifold manifold lerian lerian : : 2 1 1 Kah a a of of immersion immersion Kahlerian Kahlerian a a be be (4) (4) Opn Nr Nr Let Let ~ ~ 5.1. 5.1. Example Example 'lj; : : 1 M. M. of of connection connection Levi-Civita Levi-Civita = = the the to to respect respect with with parallel parallel is is is, is, that that holds, holds, 0 0 P P if if only only and and if if '\1 '\1 P P product product a a OR is is manifold manifold Kahlerian Kahlerian a a of of submanifold submanifold a a that that M M [6] [6] in in proved proved is is It It respectively. respectively. X, X, J J of of components components normal normal the the and and tangential tangential the the denote denote F X X F and and X X P P where where XETN, XETN, JX=PX+FX, JX=PX+FX, M, M, put put we we manifold manifold Kahlerian Kahlerian a a in in R-submanifold R-submanifold N N a a For For 0 0 M. M. of of 1_ 1_ N N manifold manifold sub real real totally totally a a and and Nr Nr submanifold submanifold Kahlerian Kahlerian a a of of product product Riemannian Riemannian 13 13 GEOMETRY GEOMETRY DIFFERENTIAL DIFFERENTIAL IN IN EMBEDDING EMBEDDING SEGRE SEGRE 14 BANG-YEN CHEN Theorem 5.2 ([6]). There do not exist OR-products in complex hyperbolic spaces other than Kiihlerian submanifolds and totally real submanifolds. OR-products N!j. x Nf in oph+p+hP(4) are obtained from the Segre em bedding as given in Example 5.1. More precisely, we have the following. Theorem 5.3 ([6]). Let N!j.xNf be a OR-product in Opm(4) with dime Nr = h and dimR N 1_ = p. Then we have (5.3) m ~ h+p+hp The equality sign of (5.3) holds if and only if the following statements hold: (a) N!j. is an open portion of0Ph(4). (b) Nf_ is a totally real submanifold. (c) The immersion is the following composition: N!j. x Nf_-. oPh(4) x OPP(4) .shp . oph+p+hp(4). Segre Imbeddmg Theorem 5.4 ([6]). Let N!j. x Nf be a OR-product in 0Pm(4). Then the squared norm of the second fundamental form satisfies (5.4) IlaW~ 4hp. The equality sign of (5.4) holds if and only if the following statements hold: (a) N!j. is an open portion of OPh(4). (b) Nf_ is a totally geodesic totally real submanifold. (c) The immersion is the following composition: N!j. X Nf totally .geodesi.c O ph ( 4) X O pP ( 4) Shp product Immersion Segre imbedding totally geodesic Kiihlerian origin. origin. the the at at centered centered Eq+l Eq+l of of Sq(1) Sq(1) hypersphere hypersphere unit unit the the into into p-manifold p-manifold Riemannian Riemannian a a of of immersion immersion isometric isometric an an is is Eq+l Eq+l C C Sq(1) Sq(1) Nj_ Nj_ : : w w and and e~ e~ --4 --4 into into h h dimension dimension complex complex of of manifold manifold Kahlerian Kahlerian a a of of immersion immersion Kahlerian Kahlerian em em a a c c is is e~ e~ Nr Nr : : z z Suppose Suppose --4 --4 Eq+ Eq+ )-space )-space . . 1 1 1 + + ( ( q q Euclidean Euclidean the the on on system system coordinate coordinate Euclidean Euclidean a a Wq) Wq) denote denote , , ...... wo, wo, ( ( Let Let {0}. {0}. =em- e~ e~ put put We We Zm}· Zm}· , , ...... {zb {zb system system coordinate coordinate complex complex Euclidean Euclidean natural natural a a with with m-space m-space Euclidean Euclidean complex complex the the em em be be Let Let 6.1. 6.1. Example Example non-constant. non-constant. is is function function ing ing warp its its if if product product CR-warped CR-warped non-trivial non-trivial a a called called is is product product CR-warped CR-warped A A function. function. warping warping the the f f denotes denotes where where N N submanifold submanifold j_, j_, real real totally totally a a and and Nr Nr submanifold submanifold holomorphic holomorphic a a of of Nj_ Nj_ XI XI Nr Nr product product warped warped the the is is it it if if product product CR-warped CR-warped a a [14] [14] in in Miscalled Miscalled manifold manifold Kaehler Kaehler a a of of CR-submanifold CR-submanifold A A f. f. functions functions warping warping non-constant non-constant with with Nj_ Nj_ submanifolds submanifolds real real totally totally and and Nr Nr submanifolds submanifolds holomorphic holomorphic of of 1 1 N N x x Nr Nr product product j_ j_ warped warped the the are are which which forms forms space space complex complex in in CR-submanifold CR-submanifold many many exist exist do do there there that that I] I] [14, [14, in in proved proved also also was was It It Nr. Nr. submanifold submanifold holomorphic holomorphic a a and and N N submanifold submanifold j_ j_ real real totally totally a a of of Nr Nr x x 1 1 N N product product j_ j_ warped warped the the locally locally is is which which manifold manifold Kahlerian Kahlerian a a in in CR-submanifold CR-submanifold a a exist exist not not does does there there that that I] I] [14, [14, in in proved proved was was It It [40]). [40]). (cf. (cf. product product warped warped the the of of function function warping warping the the called called is is f f function function + + The The P9F· P9F· gs gs g g have have = = we we Thus, Thus, TxM. TxM. E E X X vector vector tangent tangent any any for for (6.1) (6.1) that that such such structure structure Riemannian Riemannian the the with with equipped equipped x x F F B B manifold manifold = = the the is is F F xl xl B B M M product product warped warped The The F. F. x x F F B B 77: 77: Band Band --4 --4 x x F F B B --4 --4 7r: 7r: projections projections natural natural its its with with F F B B manifold manifold X X product product the the Consider Consider B. B. on on tion tion func positive positive a a be be f f let let and and respectively, respectively, gF, gF, and and gs gs metrics metrics Riemannian Riemannian with with equipped equipped dimensions dimensions positive positive of of manifolds manifolds Riemannian Riemannian two two be be F F and and B B Let Let OR-IMMERSIONS OR-IMMERSIONS SEGRE SEGRE PARTIAL PARTIAL AND AND PRODUCTS PRODUCTS CR-WARPED CR-WARPED 6. 6. 15 15 GEOMETRY GEOMETRY DIFFERENTIAL DIFFERENTIAL IN IN EMBEDDING EMBEDDING SEGRE SEGRE 16 BANG-YEN CHEN For each natural number a :::; h, we define a map: (6.2) by (6.3) Of:p(u, v) = ( wo( v)z1 (u), WI ( v)z1 (u), ... , wq( v )z1 (u), ... , wo( v)za(u), WI (v )za( u), ... , wq( v )za( u), Za+I (u), ... , Zm(u)) for u E Nr and v E Nj_. Then (6.2) induces an isometric immersion: ca . N X N -t cm+aq (6.4) hp · T f j_ from the warped product Nr XfNj_ with warping function f = J~J=I lzi(u)l 2 into cm+aq as a OR-warped product. We put When z: Nr = CZ ~ Ch and w: SP(l) ~ EP+l are the inclusion maps, the map (6.4) induces a map: (6.5) defined by (6.6) Sf:p(z, w) = ( wozb w1z1, ... , WpZI, ... , WQZa, W}Za, ... , WpZa, Za+b ... , zh) for z = (zJ, ... ,zh) E ch and w = (wo, ... ,wp) E SP(l) with ~~=ow]= 1. The warping function of c~ Xf SP(l) is given by The map Sf:P is an isometric OR-immersion which is a OR-warped product in ch+ap. We simply call such a OR-warped product in ch+ap a standard partial Segre C R-product. The standard partial Segre OR-immersion S~P is characterized by the fol lowing theorem (see [14, I]). ofSP(1). ofSP(1). portion portion open open an an is is N.1 N.1 (2.ii) (2.ii) e~. e~. of of portion portion open open an an is is Nr Nr (2.i) (2.i) have have we we then then (6.10), (6.10), of of case case equality equality the the satisfies satisfies product product CR-warped CR-warped the the If If (2) (2) }. }. + + ~(lnf) ~(lnf) 2p{IIY'(lnf)ll I I laW~ laW~ (6.10) (6.10) 2 2 cp cp satisfies satisfies of of form form fundamental fundamental second second the the of of norm norm squared squared The The (1) (1) have have we we em. em. Then Then m-space m-space Euclidean Euclidean complex complex in in product product CR-warped CR-warped a a ___.,em ___.,em be be N.1 N.1 cp: cp: Xf Xf Nr Nr Let Let 6.2. 6.2. Theorem Theorem following. following. the the by by characterized characterized is is Shp Shp CR-immersion CR-immersion Segre Segre partial partial standard standard the the one, one, than than greater greater is is When When o: o: c c EP+l. EP+l. SP(1) SP(1) E E 'Wp) 'Wp) = = ...... (wo, (wo, e~' e~' w w E E 'zh) 'zh) = = ...... (zb (zb z z (6.9) (6.9) for for ,0), ,0), ...... ,zh,0, ,zh,0, ,Z1Wp,Z2,··· ,Z1Wp,Z2,··· 1 ...... wo,ZIWI, wo,ZIWI, (z = = ,O) ,O) = = ...... (S~p(z,w),O (S~p(z,w),O cp(z,w) cp(z,w) (6.8) (6.8) have have we we precisely, precisely, More More S~p· S~p· sion sion R-immer C C Segre Segre partial partial standard standard cp cp the the em, em, is is of of motions motions rigid rigid to to Up Up (b.4) (b.4) lzii· lzii· f f = = by by given given is is function function warping warping The The (b.3) (b.3) SP(1). SP(1). p-sphere p-sphere unit unit the the of of portion portion open open an an is is N.1 N.1 (b.2) (b.2) ofe~. ofe~. portion portion open open an an is is Nr Nr (b.1) (b.1) holds: holds: statements statements following following the the if if only only and and if if 2piiY'(lnf)W 2piiY'(lnf)W = = llall equality equality the the satisfies satisfies product product 2 2 CR-warped CR-warped The The (b) (b) 2piiY'(lnf)ll ~ ~ · · 1\aW 1\aW (6.7) (6.7) 2 ity ity inequal the the satisfies satisfies form form fundamental fundamental second second the the of of norm norm squared squared The The (a) (a) have: have: we we Then Then p. p. = = N.1 N.1 d.imR d.imR and and h h = = Nr Nr dime dime em em with with m-space m-space Euclidean Euclidean complex complex the the in in product product CR-warped CR-warped non-trivial non-trivial a a ___.,em ___.,em be be :r :r cp cp XfN.l XfN.l Let Let 6.1. 6.1. Theorem Theorem 17 17 GEOMETRY GEOMETRY DIFFERENTIAL DIFFERENTIAL IN IN EMBEDDING EMBEDDING SEGRE SEGRE 18 BANG-YEN CHEN (2.iii) There exists a natural number a :::; h and a complex coordinate system { ZI, ... , zh} on Ch such that the warping function f is given by f = { tZjZj}1/2 J=1 (2.iv) Up to rigid motions ofCm, ¢is the standard partial 8egre OR-immers ion 8/:p; namely, we have (6.11) for z = (z1, ... 'zh) E c~ and w = (wo, ... 'wp) E 8P(1) c EP+ 1 . 7. REAL HYPERSURFACES AS PARTIAL SEGRE EMBEDDINGS A contact manifold is an odd-dimensional manifold M 2n+l equipped with a 1-form TJ such that TJ 1\ (dTJ)n =I 0. A curve 1 = 1(t) in a contact manifold is called a Legendre curve if TJ(f3'(t)) = 0 along j3. We put S2"+1(c) = { (z1, ... , Zn+l) E c•+l : (z, z) = ~ > 0 }· Let ~ be a unit normal vector of 8 2n+ 1 (c) in en+ 1 . Then 82n+l (c) is a contact manifold endowed with a canonical contact structure given by the dual1-form of J~, where J is the complex structure on cn+1. Legendre curves are known to play an important role in the study of contact manifolds. For instance, a diffeomorphism of a contact manifold is a contact transformation if and only if it maps Legendre curves to Legendre curves. There is a simple relationship between Legendre curves and a second order differential equation obtained in [11]. Lemma 7.1. Let c be a positive number and z = (z1, Z2) :I--+ 8 3 (c) c C 2 be a unit speed curve, where I is either an open interval or a circle. If z satisfies the following differential equation: (7.1) z"(t)- i.X.1(t)z'(t) + cz(t) = 0 manifolds manifolds of of positive positive dimension. dimension. space space forms forms which which are are the the Riemannian Riemannian products products of of two two or or more more Riemannian Riemannian In In other other words, words, there there do do not not exist exist real real hypersurfaces hypersurfaces in in non-flat non-flat complex complex in in a a complex complex hyperbolic hyperbolic space) space) is is locally locally an an irreducible irreducible Riemannian Riemannian manifold. manifold. Theorem Theorem 7.2 7.2 ([25]). ([25]). Every Every real real hypersurface hypersurface in in a a complex complex projective projective space space (or (or result result for for real real hypersurfaces hypersurfaces in in non-flat non-flat complex complex space space forms. forms. Recently, Recently, B. B. Y. Y. Chen Chen and and S. S. Maeda Maeda proved proved in in [25] [25] the the following following general general hypersurfaces hypersurfaces in in non-flat non-flat complex complex space space forms. forms. For For instance, instance, they they do do not not admit admit totally totally umbilical umbilical hypersurfaces hypersurfaces and and Einstein Einstein impose impose significant significant restrictions restrictions on on the the geometry geometry of of their their real real hypersurfaces. hypersurfaces. might might be be regarded regarded as as the the simplest simplest after after the the spaces spaces of of constant constant curvature, curvature, they they an an active active field field over over the the past past three three decades. decades. Although Although these these ambient ambient spaces spaces The The study study of of real real hypersurfaces hypersurfaces in in non-flat non-flat complex complex space space forms forms has has been been en en XC XC en en C C ofCn, ofCn, C where where Cis Cis a a real real X X curve curve in in C. C. 1 1 either either a a partial partial Segre Segre immersion immersion defined defined (7.2) (7.2) by by or or a a product product real real hypersurface: hypersurface: product product Nx1I Nx1I of of a a complex complex hypersurface hypersurface Nand Nand an an open open interval interval I, I, in in cn+l cn+l is is Conversely, Conversely, up up to to rigid rigid motions, motions, every every real real hypersurface, hypersurface, which which is is the the warped warped # # C~ C~ Zn) Zn) : : = = Zl Zl {( {( 0}. 0}. ZI, ZI, ...... , , defines defines a a CR-warped CR-warped product product real real hypersurface, hypersurface, C~ C~ I, I, in in cn+l, cn+l, Xalztl Xalztl where where (7.2) (7.2) Zn, Zn, x(zb x(zb ...... , , t) t) (af1(t)z1, (af1(t)z1, Zn), Zn), af2(t)z1, af2(t)z1, = = =j; =j; Zl Zl 0 0 Z2, Z2, ••• ••• , , I. I. Then Then the the partial partial Segre Segre immersion: immersion: a a unit unit speed speed Legendre Legendre c c curve curve 1: 1: I---+ I---+ (a S C defined defined on on an an open open interval interval ) ) 3 2 2 2 Theorem Theorem ([16]). ([16]). 7.1 7.1 Let Let 1 a a be be 2 a a positive positive number number 1(t) 1(t) and and (r (t), (t), f (t)) (t)) = = be be classification classification theorem. theorem. For For real real hypersurfaces hypersurfaces in in complex complex Euclidean Euclidean spaces, spaces, we we have have the the following following then then it it satisfies satisfies the the differential differential (7.1) (7.1) equation equation for for some some real-valued real-valued function>.. function>.. curve curve S in in (c). (c). Conversely, Conversely, if if z z z(t) z(t) is is a a = = Legendre Legendre c c curve curve S in in (c) (c) C , , 3 3 2 for for some some nonzero nonzero real-valued real-valued function function >. >. on on I, I, then then z z z(t) z(t) is is a a = = Legendre Legendre SEGRE SEGRE EMBEDDING EMBEDDING IN IN DIFFERENTIAL DIFFERENTIAL GEOMETRY GEOMETRY 19 19 20 BANG-YEN CHEN In contrast, there do exist many real hypersurfaces in non-flat complex space forms which are warped products. For real hypersurfaces in complex projective spaces, we have the following classification theorem. Theorem 7.3 ([16]). Suppose that a is a positive number and 'l'(t) = (r1(t), r2(t)) is a unit speed Legendre curve'/': I-+ S3(a2) c C2 defined on an open interval I. Let X : s~n+l X I -+ cn+2 be the map defined by the partial Segre immersion: n x(zo, ... ,Zn,t) = (arl(t)zo,ar2(t)zo,z~, ... ,Zn), L:zkzk = 1. k=O Then (1) X induces an isometric immersion '1/J : s~n+l X aizo I I -+ s 2n+3 • (2) The image '1/J(S~n+l Xalzol I) in S 2n+3 is invariant under the action of U(1). (3) the projection '1/J'Tr : 7r(S~n+l Xalzol I) -+ cpn+1(4) of '1/J via 1l' is a warped product hypersurface CP(; Xaizol I in cpn+1(4). Conversely, if a real hypersurface in cpn+1(4) is a warped product N Xf I of a complex hypersurface N of cpn+1(4) and an open interval I, then, up to rigid motions, it is locally obtained in the way described above via a partial Segre immersion. 8. COMPLEX EXTENSORS, LAGRANGIAN SUBMANIFOLDS AND SEGRE EMBEDDING When a= h = 1, the partial Segre CR-immersion (8.1) defined in Section 6 is given by (8.2) for z E C* = C- {0} and w = (wo, ... , wp) E'SP(1) C EP+l In this section, we discuss the notion of complex extensors introduced in [10] which are constructed in a way similar to (8.2). ii) ii) 11) 11) 11) 11) ii, ii, + + 11 11 + + ( ( + + ( ( {(X, {(X, Y) Y) {3 {3 ( ( JX, JX, JY JY JY, JY, JX} JX} ii) ii) ( ( ii) ii) H H (8.6) (8.6) Y) Y) O"(X, O"(X, =a =a ( ( JX, JX, JY, JY, The The condition condition (8.5) (8.5) is is equivalent equivalent to to the the single single condition: condition: local local frame frame en}· en}· field field { { e1, e1, ...... , , for for some some suitable suitable functions functions J.L J.L and and >.. >.. with with respect respect to to some some suitable suitable orthonormal orthonormal O"(ebej)=J.LJej, O"(ebej)=J.LJej, O'(ej,ek)=O, O'(ej,ek)=O, j#k, j#k, j,k=2, j,k=2, ...... ,n ,n = = = = · · · · · · = = (8.5) (8.5) A.Je1, A.Je1, O'(en, O'(en, O'(e1, O'(e1, en)= en)= e1) e1) J.LJe1, J.LJe1, O'(e2, O'(e2, e2) e2) mental mental form form takes takes the the following following form: form: ifold ifold is is a a non-totally non-totally geodesic geodesic Lagrangian Lagrangian submanifold submanifold whose whose second second funda Definition Definition 8.1. 8.1. A A Lagrangian Lagrangian H-umbilical H-umbilical submanifold submanifold of of a a Kahlerian Kahlerian man troduced troduced in in [10, [10, 11]. 11]. Now, Now, we we recall recall the the definition definition of of Lagrangian Lagrangian H-umbilical H-umbilical submanifolds submanifolds in en. en. we we can can construct construct many many S(n)-invariant S(n)-invariant Lagrangian Lagrangian submanifolds submanifolds in in e e en. en. curve curve in in is is a a SO(n)-invariant SO(n)-invariant Lagrangian Lagrangian submanifold submanifold in in this this way, way, In In sn- The The complex complex tensor tensor of of the the unit unit hypersphere hypersphere ~ ~ via via a a unit unit speed speed E 1 1 0 0 em. em. immersed immersed by by as as a a totally totally real real sub sub manifold manifold in in T T for for s s Mn- and and . . u u I I was was proved proved in in Mn- [10] [10] It It that that E E is is E E isometrically isometrically I I X X 1 1 1 u) u) (z(s)x1(u), (z(s)x1(u), (s, (s, ~----+ ~----+ ...... , , z(s)xm(u)) z(s)xm(u)) (8.4) (8.4) em em Mn- --+ --+ I I T T : : X X 1 1 is is defined defined to to be be the the map: map: e e The The complex complex extensor extensor Mn- of of x x : : --+ --+ via via Em Em the the unit unit speed speed curve curve z z : : I I --+ --+ 1 1 whose whose image image s~- (1). (1). is is contained contained in in 1 is is an an isometric isometric (n- immersion immersion of of a a Riemannian Riemannian Mn- 1)-manifold 1)-manifold En En into into 1 1 u u ,xm(u)) ,xm(u)) (x1(u), (x1(u), ...... ~----+ ~----+ c c (8.3) (8.3) (xi, (xi, ...... ' ' Mn- s~- (1) (1) Xm): Xm): --+ --+ Em Em X= X= 1 1 1 e• e• plane plane defined defined on on an an open open I. I. interval interval Suppose Suppose that that = = e• e• e e z z z(s) z(s) Let Let :I--+ :I--+ be be a a unit unit speed speed curve curve in in the the C C punctured punctured complex complex Complex Complex extensors extensors are are defined defined in in [10] [10] as as follows: follows: SEGRE SEGRE EMBEDDING EMBEDDING IN IN DIFFERENTIAL DIFFERENTIAL GEOMETRY GEOMETRY 21 21 22 BANG-YEN CHEN for vectors X, Y tangent toM, where ). + (n- 1)p, != n -t when H ol 0. It is easy to see that non-minimal Lagrangian H-umbilical submanifold satisfies the following two conditions: -t (a) J H is an eigenvector of the shape operator A H. (b) The restriction of AH to (fH)l_ is proportional to the identity map. On the other hand, since the second fundamental form of every Lagrangian submanifold satisfies (see [26]) (a(X, Y), JZ) = (a(Y, Z), JX) = (a(Z, X), JY) for vectors X, Y, Z tangent toM, we know that Lagrangian H-umbilical sub manifolds are indeed the simplest Lagrangian submanifolds which satisfy both Conditions (a) and (b). Hence, we can regard Lagrangian H-umbilical sub manifolds as the simplest Lagrangian submanifolds, next to the totally geo desic ones. Example 8.1. (Whitney's sphere). Let w: sn---+ en be the map defined by ( ) 1 + iyo ( 2 2 2 0 0 0 1 w Yo, YI, , Yn = 2 Yl, o o o , Yn), Yo + YI + o 0 0 + Yn = · 1 +Yo Then w is a (non-isometric) Lagrangian immersion of the unit n-sphere into en which is called the Whitney n-sphere. The Whitney n-sphere is a complex extensor of the inclusion t: sn-I ---+En via the unit speed curve z which is an arclength reparametrization of the curve r.p : I ---+ e given by f ( Whitney's n-sphere is a Lagrangian H-umbilical submanifold that satisfies (8.5) with A = 3p,. In fact, up to dilations, Whitney's n-sphere is the only Lagrangian H-umbilical submanifold in en satisfying A= 3p, (see [3, 10, 42]). e•. e•. in in curve curve speed speed unit unit a a via via En En of of hypersphere hypersphere unit unit the the of of extensor extensor complex complex a a Lis Lis ofen, ofen, motions motions rigid rigid to to up up then, then, curvature, curvature, sectional sectional constant constant of of subset subset (2) (2) open open no no contains contains M M If If pseudo-sphere. pseudo-sphere. Lagrangian Lagrangian a a is is L L en, en, of of motions motions rigid rigid to to up up or, or, flat flat is is M M either either then then curvature, curvature, sectional sectional constant constant of of is is M M (1) (1) If If have: have: we we Then Then isometric immersion. immersion. isometric -umbilical -umbilical H H 2: 2: Lagrangian Lagrangian a a be be en en M M : : L L and and n n 3 3 Let Let ([10]). ([10]). 8.2 8.2 Theorem Theorem ---+ ---+ 2: 2: 3. 3. with with n n en en in in submanifold submanifold H-umbilical H-umbilical Lagrangian Lagrangian classifies classifies theorem theorem following following The The constant. constant. nonzero nonzero a a is is b b case, case, this this in in Moreover, Moreover, field. field. frame frame local local orthonormal orthonormal suitable suitable some some to to respect respect with with b b function function nontrivial nontrivial some some for for , , ...... 2, 2, n, n, ek) ek) 0, 0, k k k, k, j, j, u(e1, u(e1, u(ej, u(ej, j j =/= =/= = = bJej, bJej, ei) ei) = = = = bJe1, bJe1, u(e2,e2) u(e2,e2) u(en,en) u(en,en) u(e1,el) u(e1,el) = = (8.7) (8.7) 2bJei, 2bJei, = = · · · · · · = = = = satisfying satisfying immersion immersion -umbilical -umbilical Lagrangian Lagrangian a a is is L L if if H H only only and and if if pseudo-sphere pseudo-sphere Lagrangian Lagrangian a a is is L L en, en, of of motions motions rigid rigid to to up up Then, Then, en en immersion. immersion. isometric isometric Lagrangian Lagrangian a a be be M---+ M---+ Let Let ([10]). ([10]). 8.1 8.1 L: L: Theorem Theorem following. following. the the by by characterized characterized is is pseudo-sphere pseudo-sphere Lagrangian Lagrangian The The [10]). [10]). (see (see with with (8.5) (8.5) isfying isfying A= A= 2J.L 2J.L sat submanifold submanifold H-umbilical H-umbilical Lagrangian Lagrangian a a is is pseudo-sphere pseudo-sphere Lagrangian Lagrangian A A [10]. [10]. pseudo-sphere pseudo-sphere Lagrangian Lagrangian a a as as known known is is submanifold submanifold Lagrangian Lagrangian en. en. This This into into sn(b of of ) ) 2 portion portion open open an an of of immersion immersion isometric isometric Lagrangian Lagrangian a a is is curve curve speed speed unit unit this this ---+En ---+En via via sn-l sn-l oft: oft: extensor extensor complex complex the the metric, metric, induced induced the the to to respect respect With With . . 2bi 2bi z(s) z(s) = = 1 1 + + e2bsi e2bsi e e by by given given curve curve speed speed unit unit the the be be z: z: let let R---+ R---+ > > b b 0, 0, number number real real given given a a For For pseudo-spheres). pseudo-spheres). (Lagrangian (Lagrangian 8.2. 8.2. Example Example 23 23 GEOMETRY GEOMETRY DIFFERENTIAL DIFFERENTIAL IN IN EMBEDDING EMBEDDING SEGRE SEGRE 24 BANG-YEN CHEN Remark 8.1. Flat Lagrangian H-umbilical submanifolds in en are not nec essary complex extensors (see [10]). For the explicit representation formula of flat Lagrangian H-umbilical submanifolds in en, see [12]. Remark 8.2. Complex extensors in an indefinite complex Euclidean space e~ are introduced and are investigated in [22]. For the relationship between com plex extensors and Lagrangian submanifolds in indefinite complex Euclidean spaces and their applications, see [22]. 9. PARTIAL SEGRE OR-IMMERSIONS IN COMPLEX PROJECTIVE SPACE Let C* = C- {0} and c:+I = cm+I - {0}. Consider the action of C* on c:+ 1 defined by A· (zo, ... , zm) = (Azo, ... , AZm) for A E e•, where {zo,, .. , zh} is a natural complex Euclidean coordinate system on c:+I. Let 1r(z) denote the equivalent class that contains z. Then we have a projection: 7r : c:+I - c:+I I rv • It is known that the set of equivalent classes under 1r is the complex projective m-space CPm(4). The coordinate system {zo, ... , Zm} are the homogeneous coordinate system on cpm(4). Thus, we have the projection: 1r: c:+~- cpm(4). For each integer a with 0 ~ a ~ h. We put e~+I = {(zo, ... , zh) E eh+I : t lzil 2 =/= 0}. J=O Consider the map: (9.1) defined by (9.2) Shp(z,w) = (woZO,WIZ!, ... , WpZ(), ... , WoZa, W!Za, ... , WpZa, Za+b ... , zh) for z = (zi, ... 'zh) E c~ and w = (wo, ... 'wp) E SP(1). , , · · · · · · (zowo, (zowo, 0), 0), , , ...... 0, 0, , , ...... z1, z1, zowp, zowp, zh, zh, = = 0) 0) , , = = ...... 0, 0, w), w), w) w) (S~p(z, (S~p(z, (9.6) (9.6) ¢(z, ¢(z, is, is, that that R-immersion, R-immersion, 0 0 Segre Segre partial partial standard standard s~p s~p the the is is ~P' ~P' where where ¢ ¢ composition composition the the is is (b.3) (b.3) motions, motions, rigid rigid to to Up Up 7r 7r 0 0 sP. sP. p-sphere p-sphere unit unit a a of of (b.2) (b.2) portion portion open open an an is is 1_ 1_ N N 0Ph(4). 0Ph(4). h-space h-space projective projective complex complex of of (b.l) (b.l) portion portion open open an an is is Nr Nr hold: hold: statements statements following following the the if if (9.5) (9.5) only only and and if if of of case case equality equality the the satisfies satisfies (b) (b) product product OR-warped OR-warped The The 2piiV'(lnf)ll llall ~ ~ (9.5) (9.5) • • 2 2 2 ity ity inequal the the satisfies satisfies form form fundamental fundamental second second the the (a) (a) of of norm norm squared squared The The = = = = have: have: we we Then Then 1_. 1_. N N p p and and Nr Nr h h dimR dimR where where product, product, dime dime opm(4) opm(4) OR-warped OR-warped a a be be II]). II]). N1_ N1_ ([14, ([14, Nr Nr : : 4> 4> Let Let 9.1 9.1 Theorem Theorem Xf Xf -+ -+ theorem. theorem. lowing lowing S~P S~P fol the the by by characterized characterized is is OR-immersion OR-immersion Segre Segre partial partial standard standard The The oph+p+ap(4). oph+p+ap(4). in in OR-immersion OR-immersion Segre Segre partial partial standard standard a a oph+p+aP(4) oph+p+aP(4) in in Shp Shp immersion immersion product product OR-warped OR-warped a a such such call call simply simply We We f. f. manifold. manifold. non-compact non-compact a a is is Clearly, Clearly, say say P/; P/; 0 0 function, function, warping warping with with metric metric product product warped warped a a SP(l) SP(l) is is (9.3) (9.3) via via induced induced x x OP/; OP/; on on metric metric The The ,1=0 ,1=0 t t o}. o}. o~ o~ f: f: lzil , , ...... {(zo, {(zo, OPh(4): OPh(4): E E = = zh) zh) 2 2 by by defined defined OPh(4) OPh(4) of of subset subset open open the the is is OP/; OP/; where where SP(l) SP(l) oph+p+ap(4), oph+p+ap(4), into into OP/; OP/; manifold manifold product product the the of of X X (9.4) (9.4) OR-immersion OR-immersion isometric isometric an an induces induces projection projection 4) 4) ( ( ph+p+ap ph+p+ap 0 0 1r_~ 1r_~ __ __ * * embedding embedding Segre Segre p p (9.3) (9.3) SP(l) SP(l) :~+I :~+I sh sh x x o o hp hp 1r 1r so. so. h+p+ap+I h+p+ap+I composition composition the the of of action action the the C*, C*, under under invariant invariant is is Shp Shp of of image image the the Since Since 25 25 GEOMETRY GEOMETRY DIFFERENTIAL DIFFERENTIAL IN IN EMBEDDING EMBEDDING SEGRE SEGRE 26 BANG-YEN CHEN forz = (Zo,Zl,···,zh) E e~+I and w = (wo, ... ,wp) ESP c Ep+l, and 7r is the natural projection 7r : e~+ 1 ~ c pm ( 4). The standard partial Segre CR-imrnersions Sh,p with o: > 0 are characterized by the following theorem (see [20]). Theorem 9.2. Let¢ : Nr Xf Nl_ ~ CPm(4) be a CR-warped product with h = dime Nr and p = dimR N j_. Then we have: (1) The squared norm of the second fundamental form of¢ satisfies the inequality (9.7) llull 2 ~ 2p{IIV'(lnf)ll 2 + ~(lnf)} + 4hp. (2) The CR-warped product satisfies the equality case of (9.7) if and only if the following statements hold: (2.a) Nr is an open portion of complex projective h-space CPh(4). (2.b) Nl_ is an open portion of unit p-sphere SP. (2.c) There exists a natural number o: :S h such that, up to rigid motions, ¢ is given by 1r o Sh,P, where Sh,p is the standard partial Segre CR immersion, that is, (9.8) It follows from Example 7.1 that there exist many CR-warped products Nr x f N j_ with non-constant warping function in C ph+p with h = dime Nr and p = dimRNJ_. In contrast, when Nr is compact, the following theorem shows that the dimension of the ambient space is at least as the dimension of the Segre em bedding. Theorem 9.3 ([21]). Let Nr Xf Nl_ with h = dime Nr and p = dimR Nl_ be a C R-warped product in the complex projective m-space C pm (4). If Nr is compact, then we have (9.9) m ~ h+p+hp. SEGRE EMBEDDING IN DIFFERENTIAL GEOMETRY 27 When the dimension of the ambient space C pm is m = h + p + hp which is the smallest possible, we have the following. Theorem 9.4 ([21]). Let Nr Xf N1_ with h =dime Nr and p = dimaN1_ be a CR-warped product which is embedded in cph+p+hP(4). If Nr is compact, then Nr is holomorphically isometric to CPh(4). 10. CONVOLUTION OF RIEMANNIAN MANIFOLDS The notion of a convolution of Riemannian manifolds was introduced in [18, 19]. This extends the notion of a warped product in a natural way. Definition 10.1. Let (Nb 91) and (N2, 92) be two Riemannian manifolds and let f and h be two positive differentiable functions on N1 and N2, respectively. Consider the symmetric tensor field h91 *J 92 of type (0,2) on N1 x N2 defined by (10.1) The symmetric tensor field h91 *f 92 is called the convolution of 91 and 92 via hand f. The product manifold N1 x N2 together with h91 *J 92, denoted by hN1* JN2, is called a convolution manifold. If h91 *f 92 is a positive-definite symmetric tensor, it defines a Riemannian metric on N1 X N2. In this case, h91 *f 92 is called a convolution metric and the convolution manifold hN1 *f N2 is called a convolution Riemannian manifold. When f, h are irrelevant, hN1 * f N2 and h91 *f 92 are simply denoted by N1 * N2 and 91 * 92, respectively. The following result shows that the notion of convolution manifolds arises very naturally. Theorem 10.1 ([18]). Let x: (Nb 91) -+ E~ C En andy : (N2, 92)-+ E:n C Em be isometric immersions of Riemannian manifolds (Nb 91) and (N2, 92) into E~ and E:n, respectively. Then the map (10.2) (u,v) f-+ x(u) ®y(v), 28 BANG-YEN CHEN gives rise to a convolution manifold N1 * N2 equipped with (10.3) where 2 2 PI= { txJ}l/ P2 = { f Y~}l/ J=l a=l denote the distance functions of x andy, and x=(xJ, ... ,xn), y=(Yb···,Ym) are Euclidean coordinate systems of En and Em, respectively. Definition 10.2. Let '1/J: (hNI*IN2,h9I * 192) ~ (M,g) be a map from a convolution manifold into a Riemannian manifold. Then the map is said to be isometric if h9I * !92 is induced from g via 'lj;, that is, we have (10.4) ·'··-'!-' g = h9I * /92· Example 10.1. Let x : (N1, gi) ~ E~ C En be an isometric immersion. If y : (N2,92) ~ sm-1(1) c Em is an isometric immersion such that y(N2) is contained in the unit hypersphere sm-1(1) centered at the origin. Then the convolution 91 * 92 of 91 and 92 is nothing but the warped product metric: g = 91 + lxl 292· Definition 10.3. A convolution h9I * 192 of two Riemannian metrics 91 and 92 is said to be degenerate if det(h91 *1 92) = 0 holds identically. For X E T(NI) we denote by IXh the length of X with respect to metric 91 on N1. Similarly, we denote by IZI2 for Z E T(N2) with respect to metric 92 on N2. Proposition 10.1 ([18]). Let hNI* 1N2 be the convolution of two Riemann ian manifolds ( N1, 91) and ( N2, 92) via h and f. Then h9I *1 92 is degenerate if and only if we have: (1) The length lgradfii of the gradient off on (NI,9I) is a nonzero con stant, say c. (2) The length lgradhl2 of the gradient of h on (N2,g2) is the constant given by c-1 , that is, the reciprocal of c. 7/J. 7/J. JL,~+l JL,~+l x& x& under under field field vector vector same same the the to to mapped mapped are are lxl lxl of of gradient gradient = = VL,~=l VL,~=l the the and and lzl lzl of of gradient gradient = = the the that that ZjZj ZjZj implies implies (11.3) (11.3) Equation Equation respectively. respectively. and and Ch Ch EP, EP, in in E~ E~ and and C~ C~ of of fields fields vector vector position position the the but but L,~=l L,~=l xnofoxn xnofoxn nothing nothing are are L,~=l L,~=l and and fields fields vector vector the the that that Notice Notice Zj8/8zj Zj8/8zj n=l n=l J=l J=l J J 8 8 t t t t ~n) ~n) ~.) ~.) · · Xn Xn ( ( d'ljJ d'ljJ = = ( ( d'ljJ d'ljJ Zj Zj (11.3) (11.3) that that (11.2) (11.2) from from = = obtain obtain we we then then , , ...... h, h, 1, 1, for for j j 0:,) 0:,) ~ ~ &~j &~j - i i (,::, (,::, = = .J=I, .J=I, = = = = and and put put we we If If +ivj,i +ivj,i Uj Uj Zj Zj map. map. Se9re Se9re Euclidean Euclidean a a cZ cZ called called is is (11.1) (11.1) E~. E~. xp) xp) map map The The E E x x , , ...... (xr, (xr, zh) zh) and and E E z z , , = = ...... (zr, (zr, for for = = (11.2) (11.2) by by defined defined (11.1) (11.1) 7/J 7/J map: map: the the be be Let Let maps. maps. inclusion inclusion the the cz cz EP EP are are E~ E~ and and : : z z --> --> that that Em Em Suppose Suppose : : --> --> X X on on ch ch system system 0 0 coordinate 'Xm) 'Xm) Euclidean Euclidean a a is is en en (xr, (xr, and and of of system system coordinate coordinate Euclidean Euclidean •• •• complex 0 0 'Zn) 'Zn) a a = = is is ...... (zr, (zr, that that = = Assume Assume Em- {0}. {0}. - E::' E::' en en and and {0} {0} e~ e~ Let Let MAPS MAPS SEGRE SEGRE EUCLIDEAN EUCLIDEAN AND AND CONVOLUTIONS CONVOLUTIONS 11. 11. < < 1. 1. ·lgradhl2 ·lgradhl2 lgradflr lgradflr (10.5) (10.5) * * have have we we if if only only and and if if 1 1 N2 N2 hNl hNl on on * * metric metric Riemannian Riemannian a a is is 192 192 h9l h9l Then Then f. f. and and h h via via ( ( 92) 92) N2, N2, and and ( ( 91) 91) Nr, Nr, ifolds ifolds man Riemannian Riemannian of of convolution convolution the the be be 1N2 1N2 hNl* hNl* Let Let ([18]). ([18]). 10.2 10.2 Theorem Theorem metric. metric. Riemannian Riemannian a a be be to to metrics metrics Riemannian Riemannian * * two two of of 1 1 92 92 h9l h9l convolution convolution a a for for criterion criterion a a provides provides result result following following The The 29 29 GEOMETRY GEOMETRY DIFFERENTIAL DIFFERENTIAL IN IN EMBEDDING EMBEDDING SEGRE SEGRE 30 BANG-YEN CHEN From (11.2) and (11.3) it follow that d'lj; has constant rank 2h+p-1. Hence '!j;(eZ x ~) gives rise to a (2h + p- I)-manifold, denoted by (11.4) which equips a Riemannian metric induced from the canonical metric on eh ® EP via 'If;. From (11.2) we can verify that ez ®~is isometric to the warped product ez X SP-I equipped with the warped product metric g = g1 + P1go, where PI is the length of the position function of eZ and go is the metric of the unit hypersphere sp-I ( 1). If we denote the vector field in (11.3) by V, then Vis a tangent vector field of ez ® ~ with length lxllzl. The Riemannian metric on ez ® ~ is induced from the following convolution: (11.5) Definition 11.1. An isometric map (11.6) is called a CR-map if maps each complex slice ez x { v} of ez x E~ into a complex submanifold of em and it maps each real slice { u} x E~ of ez x E~ into a totally real submanifold of em. The following two theorems characterize the Euclidean Segre maps in very simple ways. These two theorems can be regarded as the Euclidean versions of Theorem 3.4 with s = 2 and Theorem 3.5. Theorem 11.1 ([19]). Let : (eZ X E~, P2gl *Pl g2) ~ em be an isometric CR-map. Then we have: (1) m ~ hp. (2) If m = hp, then, up to rigid motions of em, is the Euclidean Segre map, that is, (11.7) following following the the prove prove also also Maeda Maeda and and Maebashi Maebashi n n to to equal equal -I. -I. constant constant is is f'f f'f of of function function curvature curvature mean mean squared squared the the that that [33] [33] in in prove prove Maeda Maeda S. S. and and Maebashi Maebashi T. T. , , ...... (ZiZj)o-:;i,j-:;n· (ZiZj)o-:;i,j-:;n· (zo, (zo, Zn) Zn) f-+ f-+ )(4); )(4); cpn(n+ : : ---+ ---+ (12.2) (12.2) f} f} CPn(2) CPn(2) 2 embedding: embedding: skew-Segre skew-Segre the the to to reduces reduces (12.2) (12.2) embedding embedding the the a:= a:= 1, 1, If If CP"'(~). CP"'(~). on on system system coordinate coordinate 0 0 homogeneous homogeneous a a , , is is ...... Zn) Zn) 0 0 (zo, (zo, =a:, =a:, and and f3i f3i = = 2:~ 2:~ o:i o:i where where , , ...... Zn) Zn) (zo, (zo, by by defined defined (12.1) (12.1) embeddings embeddings holomorphic holomorphic not not but but analytic analytic real real [35] [35] in in define define Shimizu Shimizu Y. Y. and and Maeda Maeda S. S. embeddings, embeddings, Veronese Veronese the the and and Segre Segre the the by by Motivated Motivated EMBEDDING EMBEDDING SEGRE SEGRE SKEW SKEW 12. 12. 0) 0) = = ('1flz,x' ('1flz,x' x) x) cp(z, cp(z, (11.9) (11.9) by by given given is is cp cp is, is, that that map, map, Segre Segre Euclidean Euclidean the the from from em, em, obtained obtained is is cp cp of of tions tions mo rigid rigid to to up up if, if, only only and and if if identically identically holds holds (11.8) (11.8) of of sign sign equality equality The The lzl lxl ~ ~ • • u u 2 2 2 (11.8) (11.8) 11 II II 1) 1) 1)(p- (2h- 2 2 have: have: we we Then Then CR-map. CR-map. em em isometric isometric an an be be g2) g2) *Pl *Pl ---+ ---+ E~, E~, P2gl P2gl (e~ (e~ : : cp cp Let Let X X ([19]). ([19]). 11.2 11.2 Theorem Theorem 31 31 GEOMETRY GEOMETRY DIFFERENTIAL DIFFERENTIAL IN IN EMBEDDING EMBEDDING SEGRE SEGRE 32 BANG-YEN CHEN Theorem 12.1 ([33]). The skew-Segre embedding f1 is equal to the following composition: CPn(2) minimal sn(n+2)-1 (n ~ 1) totally umbilical sn(n+2)(1) totally geodesic cpn(n+2)(4). The first part, cpn(2) minimal sn(n+2)-I (n~I)' of the decomposition for the skew-Segre embedding has already been considered by G. Mannoury (1867-1956) in 1899 (see [36]) The skew-Segre embedding f1 : cpn(2) ~ cpnCn+2>(4) is a totally real pseudo-umbilical embedding which has parallel mean curvature vector. More over, the squared norm of the second fundamental form of the skew-Segre embedding is constant. Foro:= 2, the embedding (12.1) reduces to (12.3) f'f : cpn(1) ~ cp(nt2)-1(4); (Z() · · z ) ( · · · z?z2k · · · z?zkzz · · · ,. ziz2k · · · f2,, z ·zkzz · · ·) ' '- n ~ ' ' ' ' ~ ' ' "'"' ' ' y ~"'"' 'J ' • S. Maeda and Y. Shimizu gave in [35] an analogous decomposition for this embedding. Theorem 12.2 ([35]). The embedding fr: cpn(1) ~ cp(nt2)-t(4) is equal to the following composition: (12.4) CPn(1) minimal sm1 -I(ct) x sm2-I(C2) natural Clifford embedding gml+m2-l(c) totally umbilical sml+m2(1) totally real cp(nt2)-1(4), totally geodesic where 1 m1 = n(n + 2), m2 = 4n(n + 1)2(n + 4), (n + 1)(n + 3) (n + 2)(n + 3) Ct= 4n ,c2 = n(n+1) ' 2 c= 1 +n(n+3)" EN. EN. v v EM, EM, u u h(v) h(v) EB EB v) v) f(u) f(u) h)(u, h)(u, E8 E8 (f (f = = (13.8) (13.8) ®h(v), ®h(v), f(u) f(u) (JLE:Jh)(u,v) (JLE:Jh)(u,v) = = 7) 7) (13. (13. map map sum sum box-direct box-direct the the and and map map product product box-tensor box-tensor the the have have we we Then Then respectively. respectively. W, W, and and V V into into N N and and M M manifolds manifolds Riemannian Riemannian two two from from maps maps : : f f are are W W : : N N f f and and V V M M if if -4 -4 Similarly, Similarly, -4 -4 EM,. EM,. u u EB EB E E f2(u) f2(u) W, W, V V EB EB fi(u) fi(u) h)(u) h)(u) (fi (fi EB EB = = (13.6) (13.6) ® ® W, W, V V h(u) h(u) fi(u) fi(u) ® ® E E h)(u) h)(u) (!I® (!I® = = (13.5) (13.5) by by defined defined are are maps maps g). g). (M, (M, These These manifold manifold Riemannian Riemannian : : a a h h of of W W M M and and : : fi fi V V M M -4 -4 maps maps given given two two with with -4 -4 associated associated VEBW VEBW M M EBh: EBh: fi fi -4 -4 (13.4) (13.4) fi®f2:M-4V®W fi®f2:M-4V®W (13.3) (13.3) maps maps sum sum direct direct and and product product tensor tensor of of notion notion the the have have we we notions, notions, algebraic algebraic these these applying applying By By vEBw,xEByEVEBW vEBw,xEByEVEBW =(v,x)v+(w,y)w, =(v,x)v+(w,y)w, (vEBw,xEBy) (vEBw,xEBy) (13.2) (13.2) V®W, V®W, E E v®w,x®y v®w,x®y = = (w,y)w, (w,y)w, · · (v,x)v (v,x)v (v®w,x®y) (v®w,x®y) (13.1) (13.1) by by respectively respectively defined defined products products inner inner the the with with spaces spaces product product inner inner are are EB EB W W V V and and W W ® ® V V Then Then respectively. respectively. W W and and on on V V products products inner inner the the denote denote )w )w (, (, and and )v )v Let(, Let(, respectively. respectively. and and W, W, V V of of sum sum direct direct the the and and product product tensor tensor the the EB EB W W V V and and W W V V by by ® ® Denote Denote numbers. numbers. complex complex or or real real of of field field the the over over spaces spaces vector vector two two W W be be V V and and Let Let details). details). for for 28] 28] [9, [9, (see (see maps maps sum sum direct direct and and maps maps product product tensor tensor of of notions notions the the recall recall we we Here, Here, map. map. product product tensor tensor a a as as regarded regarded be be can can embedding embedding Segre Segre the the defines defines which which (1.5) (1.5) map map The The EMBEDDING EMBEDDING SEGRE SEGRE AND AND IMMERSIONS IMMERSIONS PRODUCT PRODUCT TENSOR TENSOR 13. 13. constant. constant. is is curvature curvature mean mean the the Jr Jr although although bundle bundle normal normal the the in in parallel not not is is of of vector vector curvature curvature mean mean the the and and Jr Jr embedding embedding real real totally totally pseudo-umbilical pseudo-umbilical a a is is that that observe observe They They 33 33 GEOMETRY GEOMETRY DIFFERENTIAL DIFFERENTIAL IN IN EMBEDDING EMBEDDING SEGRE SEGRE 34 BANG-YEN CHEN The Segre embedding, the partial Segre immersions, complex extensors, Euclidean Segre maps, convolutions, as well as skew-Segre immersions given above can all be expressed in terms of (box) tensor product maps and (box) direct sum maps. Example 13.1. Let 1.1 : ez- eh and 1.2: e~- cP be the inclusion maps. Then the map Shp defined by (1.5) is nothing but the box tensor product 1.1 il:l1.2 of 1.1 and 1.2. Example 13.2. Let z = (zb ... , Zm) : Nr - e:- c em be a Kahlerian immersion of a Kahlerian h-manifold into e~ and w = (wo, ... , wp): N_1_- Sq(l) C Eq+l be an isometric immersion from a Riemannian p-manifold into the unit hyper sphere Sq(l). Then the partial Segre OR-immersion Chp defined by (6.3) is nothing but the map: (13.9) Ch'P = (za IZI w) EE zl_ : Nr x N_1_ ----+ em+aq (u,v) f-+ (za(u) ®w(v)) Eazl_(u), where and Example 13.3. Let z = z(s) : I- e• C e be a unit speed curve in the punctured complex plane e• defined on an open interval I and let X= (x1, ... , Xm) : Mn-1- s;;"- 1 C Em; u f-+ (x1(u), ... , Xm(u)) be an isometric immersion of a Riemannian (n- !)-manifold Mn- 1 into En whose image is contained in the unit hypersphere. Then the complex extensor T of x via the unit speed curve z is nothing but the box tensor product z ll:l x. Example 13.4. Let z = (zb ... 'Zn) : e~ - en be the inclusion map of e~ and let Z- = (-Z1, ••. , Zn- ) : en* en- be the conjugation of z : e~ -en. Then the map (12.1) which defines the skew-Segre embedding is nothing but the box tensor product z ll:l z. (14.5) (14.5) of of the the general general inequality: inequality: space. space. Such Such standard standard partial partial Segre Segre OR-immersions OR-immersions satisfy satisfy the the equality equality case case gives gives rise rise to to the the standard standard partial partial Segre Segre OR-immersion OR-immersion in in a a complex complex projective projective ohp: ohp: (14.4) (14.4) Nj_- Sq(1)- Nr Nr em+aq em+aq em em X X X X maps. maps. Then Then the the partial partial Segre Segre map: map: EP+ Example Example 14.3. 14.3. Let Let z z : : CZ CZ Ch Ch and and sP(1) sP(1) be be the the inclusion inclusion w w : : <...... + <...... + <...... + <...... + 1 1 amples amples of of OR-warped OR-warped products products in in complex complex Euclidean Euclidean spaces. spaces. gives gives rise rise to to a a OR-submanifold. OR-submanifold. Such Such construction construction provide provide us us many many nice nice ex (14.3) (14.3) defined defined by by ohp: ohp: (14.2) (14.2) Nj_- Sq(1)- Nr Nr em+aq em+aq em em X X X X = = a: a: with with h h dime dime h h the the partial partial Segre Segre Nr, ~ ~ map: map: given given Riemannian Riemannian immersion immersion Sq(1) Sq(1) w: w: Eq+l, Eq+l, N1_-+ N1_-+ and and a a natural natural number number <...... + <...... + Example Example 14.2. 14.2. For For a a given given Kahlerian Kahlerian immersion immersion z z : : Nr-+ Nr-+ a a C~ C~ <...... +em, <...... +em, OPh+p+hP(4). OPh+p+hP(4). x0.PP(4) x0.PP(4) into into gives gives Kiihlerian Kiihlerian rise rise to to a a immersion immersion of of the the product product Kahlerian Kahlerian 0Ph(4) 0Ph(4) manifold manifold (14.1) (14.1) Segre Segre map: map: Example Example 14.1. 14.1. For For the the inclusion inclusion maps maps z: z: CZ CZ cP, cP, Chand Chand ~ ~ w: w: the the c...... ,. c...... ,. <...... + <...... + previous previous sections sections that that such such examples examples have have many many nice nice properties. properties. various various important important classes classes of of submanifolds. submanifolds. Moreover, Moreover, we we also also see see from from the the in in ways ways similar similar to to the the Segre Segre embedding embedding provide provide us us many many nice nice examples examples for for From From the the previous previous sections sections we we know know that that maps maps and and immersions immersions constructed constructed 14. 14. CONCLUSION CONCLUSION is is nothing nothing but but the the first first standard standard immersion immersion of of sn(1) sn(1) (see (see [8]). [8]). (13.10) (13.10) sn(1)-+ sn(1)-+ tn tn tn tn ® ® : : E(n+1) 2 2 dilations, dilations, the the tensor tensor product product immersion: immersion: Example Example 13.5. 13.5. Let Let -+ -+ tn tn : : sn(1) sn(1) En+l En+l be be the the inclusion inclusion map. map. Then, Then, up up to to SEGRE SEGRE EMBEDDING EMBEDDING IN IN DIFFERENTIAL DIFFERENTIAL GEOMETRY GEOMETRY 35 35 36 BANG-YEN CHEN Example 14.4. For a unit speed curve z = z(s): I---> e* <...... t e and a spher ical isometric immersion X : Mn- 1 ---> sg<- 1 <...... t Em' the complex extensor T of x via z is nothing but the box tensor product z IZI x. Such box tensor prod uct immersions provide us many nice examples of Lagrangian submanifolds in complex Euclidean spaces. Example 14.5. Let a be a positive number and 'Y(t) = (r1(t),r2(t)) be a unit speed Legendre curve 1: I---> S 3 (a2) <...... t e 2 defined on an open interval I. Then the partial Segre immersion defined by (14.6) defines a warped product real hypersurface in en+1. Conversely, up to rigid motions, every real hypersurface in en+ 1 which is the warped product N x f I of a complex hypersurface N and an open interval I is either the partial Segre immersion given by (14.6) or the product real hypersurface: en X C in en+ 1 over a curve C in the complex plane. Example 14.6. Let X: (N1,g1)---> E~ <...... tEn, Y: (N2,g2)---> E:n <...... t Em be two isometric immersions of Riemannian manifolds (N1, g1) and (N2, g2) into E~ and E:n, respectively. Then the Euclidean Segre map: (14.7) X IZJ Y: N1 X N2---> En® Em; (u,v) f-4 x(u) ®y(v), gives rise to the convolution manifold N1 * N2 equipped with the convolution: (14.8) p2 g1 *p1 g2 = p~g1 + p~g2 + 2p1p2dp1 ® dp2, where p1 = lxl and p2 = IYI are the distance functions of x andy, respectively. Example 14.7. Let z : e~ <...... t en be the inclusion map of e~ and let z be the conjugation of z : e~ ---> en. Then the skew-Segre map z IZI z gives rise to a totally real isometric immersion of cpn(2) in cpn(n+2)(4). Example 14.8. Let 7./J : M ---> CPm(2) be a Kahlerian immersion from a Kahlerian manifold Minto CPm(2) and let ;j; be the conjugation of 7./J: M---> cpm(2). Then the map: (14.9) (to (to appear). appear). [20] [20] B. B. Y. Y. Chen, Chen, Another Another general general inequality inequality for for CR-warped CR-warped products products in in complex complex space space forms, forms, [19] [19] Y. Y. B. B. 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