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Generic Submanifolds (*) Generic Submanifolds (*). Kv,~TARO YA~o - MASAmUO Ko~ (Tokyo, Japan) Summary. - in this paper we give some examples of generic submanifolds of complex space forms and prove some theorems which give the characterizations of these examples..For this purpose we study the ~'elations between a submanifotd of a K~hlerian manifold and a submauifold of a Sasalcian manifold by using the method of Riemannian fibre br O. - Introduction. The theory of submanifolds of K~hlerian manifolds is parhaps one of the most interesting topics in modern differential geometry. A submanifold M of a Ki~hlerian manifold M is called a complex submanifold (an invariant submanifold) if the tangent space T,(M) of M at x is always mapped into itself under the action of the almost complex structm'e tensor J of the ambient manifold ~r, that is, if JT=(M)c T~(M) for all x ~ M. Complex submanifolds of K~hlerian manifolds have been studied by various authors and still suggest many interesting problems. On the other hand, a submanifold M of a K~hlerian manifold M is said to be totally real (anti-invariant) if JT~(M) c T~(M) • for all x ~ M, where T~(M) • denotes the normal space of M at x (see [1], [15] and [21]). Now, a submanifold M of a Ki~hlerian manifold M is called a generic submanifold (an anti-holomorphie snbmanifold) if JT.(M) • c T~(M) for all x e M (see [9] and [15]). For example, any real hypersm'face of a K~hlerian manifold is a generic submanifold. For real hypersurfaces, see [5], [7], [8], [12] and [13]. In [6], Lawson studied submanifolds of complex space forms by using the method of ;[~iemannian fibre bundles and MAEDA [7] and OKIJ~rU~A [8], [9] developed this method of Lawson extensively (see also [21]). The pal'pose of the present paper is to study generic submanifolds immersed in complex space forms by the method of l~iemannian fibre bundles. In w 1, we first of all recall some of fundamental formulas in the theory of sub- manifolds of a Riemannian manifold, give next a summary of known results and prove theorems on submanifolds of a sphere which will be used in the sequel. In w 2, we recall some of fundamental formulas in the theory of submanifolds of K~hlerian manifolds, especially in that of generic sllbmanifolds of Ki~hlerian (*) Elltr~ta in Redazione il 30 agosto 1978. 60 KE~-~A]~o u - 5IASAm~O K0g: Generic submaniJolds manifolds. Moreover, we show that any generic submanifold of a Kiihlerian manifold admits an J-structure [16] and study its integrability conditions [2]. In w 3 we give examples of generic submanifolds of a complex projective space CP~ and those of generic submanifolds of a complex number space C% w 4 is devoted to the study of relations between a submanifold M immersed in a Ki~hlerian manifold M and a submanifold h r immersed in a Sasakian manifold by using the method of Riemannian fibre bundles. We prove some lemmas giving the relations between the second fundamental form B of M and the second funda- mental form e of N. We also study relations between the normal connection of M and that of h r and those between the mean curvature vector of M and that of 2;r. Our main theorems will be proved in w 5 which give the characterizations of examples given in w3. We study in w 6 pseudo-umbflicM generic submanifolds of complex space forms. The notion of pseudo-umbilical generic submanifolds is a generalization of the notion of ~-~mbilieal real hypersarfaee. In the last w 7, we give a generalization of a theorem on anti-iuvariant submani- folds proved in [19] and an application of a theorem of Wells [15]. 1. - Submanifolds of Riemannian manifolds. Let 217 be an m-dimensionM l%iemannian manifold with positive definite metric tensor field g and 21/be an n-dimensional submanifold isometrically immersed in 2ft. We denote by the same g the !~iemannian metric tensor field induced on M. The operator of covariant differentiation with respect to the Levi-Civita connection in 2~ (resp. M) wilI be denoted by V (resp. V). Then the Gauss and Weingarten formulas are respectively given by vZg = vx:g + B(X, Y), VxF = - A~X + 1)~V for any vector fields X, Y tangent to M and any vector field V normal to M, where D denotes the operator of covariant differentiation with respect to the linear con- nection induced in the normal bundle of M. A and B are both called the second fundamental form of M and are related by g(B(X, Y), V)--= g(AvX, Y). The second fundamental form B is a vector bundle valued symmetric bilinear form on each t~ngent space T~(M) taking values in the normal space T~(M) • and the second fundamental form A is a cross-section of a vector bundle ttom (T(M) • S(M)) where S(M) denotes the bundle whose fibre at each point is the space of sym- metric linear transformations: T~(M) --->T~(M), i.e., for V E T~(M) • Av: T~(M)--~ KE~TAI~O YAI~O - 1V[ASAHIt~O KOI~I: Generic submani]o~ds 61 l%r the second fundamental form B we define its covariant derivative VzB by (VxB)(~, Z) = Dx(B(]~, Z)) -- B(Vx~, Z)- B(](, VxZ) for any vector fields X, Y and Z tangent to M. If VxB = 0 for all X, then the second fundamental form B of M is said to be parallel. This is equivalent to VxA = 0, where VxA is defined by (VxA)v 1r : Vx(Av Y) -- A,~v Y -- Av VxY . A normal vector field V on M is said to be parallel if -DxV ~- 0 for any vector field X tangent to M. The mean curvature vector # is defined to be/~ ----- (Tr B)/n, where Tr B is the ~traee of B. If B----0 identically, then M is said to be totally geodesic and if # = 0~ then M is said to be minimal. If the second fundamental form B is of the form B(X, Y) = g(X~ Y)[2~ then M is said to be totally umbilical. Let/~ and R be the Riemannian curvature tensor fields of M and M respectively. Then, for any vector fields X, ]r and Z tangent to M, we have (1.1) R(X, :g) Z =-I~(X, Y) Z -- ABr + A~r + (VxB)(:Y, Z) -- (V~B)(X, Z). From (1.1), we have equation of Gauss (1.2) g(/~(X, Y)Z, W) = g(R(X, Y)Z, W)- g(B(X, W), B(Y, Z)) + + g(B(Y, W), B(X, Z)) for any tangent vector fields X~ Y, Z and W. Taking the normal component of (1.1), we have equation of Codazzi (1.3) (_~(X, Y)Z)~= (VxB)(]~, Z)- (V~B)(X, Z). We now define the curvature tensor /~• of the normal bundle of M by R'(X, Y) V =- DxD~ V -- DyDx V -- Drx,~l V, X, :Y being vector fields tangent to M and V a vector field normal to M. Then we have equation of l~icci (1.4) giR(X, Y) U,y) = g(R~(X, Y) U, V) + g([A~, A~]X, ~) for any vector fields X~ Y tangent to M and for any vector fields U, V normal to M, where [Av~ A~] = ArAb-- A~Av. 62 KE~A~O Y~o - ~V~ASA~O Koch: Generic submani/olds If R • vanishes identically, then the normal connection of M is said to be fiat (or trivial). If, for any vector fields U and V normM to M, lAy Av] : 0, then the second fundamental form of M is said to be commutative. In the following we prepare some lemmas on submanifolds of a sphere for later use. Let M be an n-dimensionM minimal submanifold isometrically immersed in a unit sphere S ~+~ of dimension n-~ p. We denote by S the square os the length of the second fundamental form of M, that is, we put S ~- ~g(B(e~, ej), B(e~, ej)) = ~g(A~ei, A~e~) = ~TrA~, where {e~} denotes an orthonormM frame for T~(M) and {v~} denotes an orthonormM frame for T~(M)• writing A~ in place of A~ to simplify the notation. In the equa- tion above and in the sequel, the indices i, ],/% ... run over the range {1, 2, ..., n} and the indices a~ b, c, ..., the range (1, 2, ..., p}. Then the formula of Simons is given by (el. S~o~s [ii]) (z.5) AS = nS -- ~ (Tr A~Ab) ~ -p ~ Tr [A~, Ab] ~ -~ g(VA, VA), a,b a,b where g(VA, VA) = ~ g((V~A)~ej, (V~A)~ej). If the second fundamental form of M a~$~ is parallel, then S is constant and (1.5) implies (~.6) 0 < -- ~ Tr [A~, Ad ~ = nN-- ~ (Tr A~A~) ~ . a,b a,b If we put Sa~= TrA~Ab and 8~= ~o, then (S~b) is a symmetric (p, p)-matrix and so can be assumed to be diagonal for a suitable choice of v,, ..., %. Thus we have, from (1.6), (z.;) ~b a =nS ~_1 ~(S_S~)~ <1 On the other hand, from (1.4), we see that the normal connection of M is fiat if and only if the second fundamental form of M is commutative. Therefore (1.7) implies LE~:~ 1.1. - Let M be an n-dimensional minimal submani]old o] a unit sphere $'~+~ with parallel second ]undamental ]orm.
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