TOTALLY REAL SUBMANIFOLDS OF A COMPLEX SPACE FORM

Totally real submanifolds of a complex space form are studied. In particular, totally real submanifolds of a complex number space with parallel mean curvature vector are classified.

I and g(JX, JY) g(X, Y) for any vector fields.X and Y on/r, where I denotes the identity transformation on the tangent bundle.Let be the Levi-Civita connection of M satisfying J O. Let M be an n-dimensional Riemannian manifold isometrically immersed in M by the immersion i: M --M.We then obtain the induced metric on M which will be represented the same notation g.We also identify X with i.(X) and M with i(M).
Let be the induced Levi-Civita connection on M. Then the equations of Gauss and Weingarten are respectively given by x Y V x Y + h(X, Y) and x AX + V , where h is the second fundamental form, A the Weingarten map associated to the normal vector field ( satisfying g(h(X, Y),) g(AX, Y) and 7 +/-the connection in the normal bundle T+/-M of M. The mean curvature vector H is then given by H 1 Trh.An n-dimensional submanifold M in a Kaehler manifold//is called totally real if J (TpM) c T M for each P in M, where TpM is the tangent space of M at P and T,M the normal space of M at P.
Since J has the maximal rank, m _> n.Let Np(M) be the orthogonal complement of J(TpM) in TM.Then we get the decomposition TM J(TpM) Np(M).It follows that the space Np(M) is invariant under the action of J.We now consider an m-dimensional totally real submanifold M of 2m-dimensional Kaehler manifold M. Then we may set JX O(X), where X is a vector field tangent to M, 0(X) a normal vector valued 1-form, a normal vector field and U a vector field on M satisfying g(U, X) g(0(X), ).Applying J to (1. l) and (1.2), we have X Uo(x) and O(U) .
(1.3) Differentiating (1 1) and (1.2) covariantly and making use of the equations of Gauss and Weingarten, we get Uh(X,Y) Ao(x)Y, O(AX) h(X, U), (1.7) where X and Y are vector fields tangent to M and a vector field normal to M.
We now assume that the ambient manifold M is of constant holomorphic sectional curvature 4c, which is called a complex space form and it is denoted by M(c).Then the Riemann Christoffel curvature tensor R of M(c) has the form gCt(x, )z, w) ((x, w)(, z) (Y, w)(x, z) + (sx, w)(JY, z) g(JY, W)g(JX, Z) 2g(JX, Y)g(JZ, W)).
Since the manifold M is totally real, it follows from equations(1, l)-(1.7)that the equations of Gauss, Codazzi and Ricci for M are respectively obtained whereis the covariant derivative on T(M)T+/-(M) defined by (-xh)(Y,Z)= ch(Y,Z) -h( ,xY, Z)-h(Y, x Z),R and R +/-are the Riemann curvature tensor of M and that in the normal bundle respectively and [A, An]  2. FUNDAMENTAL LEMMAS.
In this section, we assume that M is an m-dimensional totally real submanifold of a complex space form M(c) of real dimension 2m A normal vector field ( is said to be parallel if 7 -( 0 for any vector field X on M and ( is called an moperlmemc section if Tr A is non-zero constant LEMMA 1.Let M be an m-dimensional totally real submanifold of M(c) with parallel isoperimetric section ( If A has no simple eigenvalues, then M (c) is flat PROOF.Since A is self-adjoint with respect to g, there exists an orthonormal basis {et, e2, , e,} for TpM such that g(Ae,, e,) A6, 3, where A1, )2, , Am are eigenvalues of A.
Choosing q as O(e,), we get cg(O(e,),() 0 By (1 1), we see that {O(e,) ,m} fos onhonofl basis for TM.It follows that M(c) is flat. (Q.E.D.) 1. Let M be an m-dimensional totally real submanifold of M(c)(c 0).If M has an isopefimetfic section , then A has simple eigenvalues Let H be the me cuature vector field defined by H Trh. We now assume that H is nonvsng pallel in the nodal bundle.We choose an onhonoal am {, 2, ,} in the nofl bundle in such a way that (1 H/ H It follows that TrA, 0 for 2, where A, A, and U, U2, U fo onhonofl basis for TpM because of (1.2), where U, U,.Then (1.3)   d (1.4) imply AtU Uh(u,,u,), ( wMch shows that A,U AU,.
Tng the scM product th d mng use of (1.3), (1.7) and (2.1), we may set A, Ua #Ua, ( k where # g(O(AUa), ).Because A, is a setfic operator d h is a syetfic bilin fo, # is setfic th respect to Ml indices i, j d k.
Let M be an m-dimensional totally real submanifold of a complex space form M(c) with nonvanishing parallel mean curvature vector.By lemrna 2, we know that AH is parallel.We now define a function hn for any integer n > 1 by h, Tr(A).Then h, is constant on M for any integer n since AH is parallel.This implies that each eigenvalue , of An is constant on M. Let #1,/2, ",/o be mutually distinct eigenvalues of AH and hi, r, , no their multiplicities.So the smooth distributions Ta consisting of all eigenvectors corresponding to #a are defined and orthogonal each other.Since AH is parallel, Ta are parallel and completely integrable.By the de Rham decomposition theorem [4], the submanifold M is a product manifold M Mg.
M,, where the tangent bundle of Ma corresponds to Ta.We now assume that the ambient manifold is fiat, that is, a complex number space C and M is embedded in C'.Then as in [1] we can choose an orthonormal basis e, e, -, e for TM as eigenvectors of AH and J,J, ., J for J(TpM) in such a way that ' o(A4 e) and hi hjk ho,, where hs/ e,, h,,/= 0 for eo E Lu3], e, E 'r],/3 :/= 7, where O] is the eigenspace corresponding to the eigenvalue Let 7re(H be the component of H in the subspace Cre.Then 7re(H is a parallel normal section of M e in C ''e and M e is umbilical with respect to 7re(H).Therefore, M e is a minimal submanifold of a hypersphere in C"e.Hence M is a product submanifold M1 x M2 x x M, embedded in Cr, C "1 C C", where M e is a totally real submanifold embedded in some Cwe.Thus we have THEOREM 1.Let M be an m-dimensional complete totally real submanifold embedded in a complex number space Cm.If M has parallel mean curvature vector H, then M is either a minimal submanifold or a product submanifold MI M2 x x M embedded in C n C 01 x C 'e2 x x Cw, where M e is a totally real submanifold embedded in some C is also a minimal submanifold of a hypersphere of C ve THEOREM 2. Let M be an m-dimensional complete totally real submanifold embedded in a complex number space Cr".If M has the nonvanishing parallel mean curvature vector and An has mutually distinct eigenvalues, then M is a product submanifold of circles S x S S1.
PROOF.By a lemma of Moore [5], M M1 x M2 Mm is a product immersion embedded in C'', and M, is a totally real submanifold in C and contained in a hypersphere in Sim;e n + rv2 + + r, m, r, must be 1.Hence M, S1, a circle in a complex space C. (Q.E.D.)

(2. 5 )
for all vector fields X and Y on M. (2.5) together with (2.3) yields and hence where P Pll1.