J. Korean Math. Soc. 32 (1995), No. 2, pp. 161-170
CHARACTERIZATIONS OF SOME REAL HYPERSURFACES IN A COMPLEX SPACE
FORM IN TERMS OF LIE DERIVATIVE*
V-HANG KI AND YOUNG JIN SUH
1. Introduction
A complexn(~2)-dimensionalKa.ehlerian manifold of constant holo- morphic sectional curvaturecis called a complex space fonn, which is denoted by Mn(c). A complete and simply connected complex space form is a complex projective space PRC, a complex Euclidean space
cn or a complex hyperbolic space HnC, according as c > 0, c = 0 or c < O. Takagi [12] and Berndt [2] classified all homogeneous real hypersurfaces ofPnC and HnC.
Now, let M be a real hypersurface of Mn(c),c;l:O. Then M has an almost contact metric structure (<p,~,"7,g) induced from the Kaehler metric and the almost complex structure ofMn(c). We denote by £E the Lie derivative with respect to e.
Recently Ki,Kim and Lee [4] gives a characterization of real hy- persurfaces of type A. We denote by A a shape operator of the real hypersurface M. They proved the following.
THEOREM A. Let M be a real hypersurface of PRG, n~3. If it satisfies
(1.1) £eA= 0,
where A denotes the shape operator, then M is locally a tube of radius
r over one of the following Kahler submanifolds:
(Ad a hyperplane Pn-IG, where 0< r< 71"/2,
Received August 12, 1993.
1991 AMS Subject Classification: 53C15, 53C45.
Key words: real hypersurfaces of typeA,ruled real hypersurface, Lie derivative.
*Partially supported by TGRC-KOSEF and BSRI94-1404.
(A2) a totally geodesicPkC (1< k < n - 2),where°< r <7r/2.
As an example of special real· hypersurfaces ofPnC different from the above ones, we can give some characterizations of ruled real hy- persurfaces in terms of the Lie derivative of the second fundmental form.
On the other hand, Kimura [7] obtained some properties about a ruled real hypersurface M of PnC,n~3. In particular, an example of minimal ruled hypersurface M ofPnC,n~3is constructed. Let To be a distribution defined by a subspace To(x) = {uETxM : u.LeCx)} of the tangent spaceTx(M),which is called theholomorphic distribution.
Kimura and Maeda [8] also proved the following
THEOREM B. Let M beareal hypeisurface ofPnC, n~3. Tben the second fundamentalform is TJ-parallel and the holomorpbic distribution To is integrableifandonlyifM 'is locally a ruled real bypersurface.
The purpose of this article is to generalize Therem A slightly and then to give another characterization of the ruled real hypersurfaces in Mn (c). Assume thateis not necessary principal. Then we can put
Ae= ne+{3U, where Uis a unit vector orthogonal to·eand a and e
are smooth functions on M. We prove the following.
THEOREM 1. Let M bea realhypersurfaceofMn(c), c,t:O, n~3.
(1.2) g((£eA)X, Y) = °
for anyvector fields X and Y in the distributionTo, then M is of type.
A.
THEOREM 2. Let M beareal hypersurface of Mn (c), ci=O andn~3.
Hit satisfies
(1.3) g((£eA)X, Y) = {32g(X, r/JU)g(Y, U)
for anyvector fields X and Y in the distributionTo andifthe structure vector field iSnotprmCip8J anddQ(erFO, then M is locally' congruent to a ruled real hypersurface.
Real hypersurfaces in terms of Lie derivative
2. Preliminaries
163
First of all, we recall fundamental properties of real hypersurfaces of a complex space form. Let M be a real hypersurface of a complex n-dimensional complex space form Mn(c) of constant holomorphic sec- tional curvature c(:FO) and let C be a unit normal field on a neighbor- hood of a point x in M. We denote by J an almost complex structure of Mn(c). For a local vector field X on a neighborhood ofx in M, the transformation ofX and Cunder J can be represented as
JX = 4>X+Tf(X)C, JC= :-e,
where 4> defines a skew-symmetric transformation on the tangent bun- dle TM of M, while Tf and edenote a I-form and a vector field on a neighborhood of x in M, respectively. Moreover, it is seen that gCe, X) = Tf(X),where 9denotes the induced Riemannian metric onM.
By properties of the almost complex structure J, the set(4),e,1],g) of
tensors satisfies
whereI denotes the identity transformation. Accordingly, the set is so called an almost contact metric structure. Furthermore the covanant derivative of the structure tensors are given by
(2.1) (Vx4»Y=Tf(Y)AX -g(AX,Y)e, Vxe =4>AX,
where V is the Riemannian connection of9 and A denotes the shape operator with respect to the unit normal Con M.
Since the ambient space is of constant holomorphic sectional cur- vature c, the equation of Gauss and Codazzi are respectively given as follows
(2.2)
R(X, Y)Z ='4c{g(Y, Z)X - g(X, Z)Y+g(4)Y, Z)4>X - g(4)X, Z)4>Y - 2g(4)X, Y)4>Z} +g(AY,Z)AX - g(AX, Z)AY,
(2.3) (VxA)Y - (VyA)X = '4c{Tf(X)4>Y - Tf(Y)4>X - 2g(4)X, Y)e},
where R denotes the Riemannian curvature tensor of M and VxA
denotes the covariant derivative of the sha.pe operatorA with respect toX.
The second fundamental form is said to be 7]-parallelif the shape operator A satisfiesg«VxA)Y,Z)~ 0fOFany vector fields X,Y and
Z inTo. .
3. Proof of Theorem 1
Let M be a real hypersu,rface of Mn(c), C:f;O,n~3. Inorder to prove Theorem 1 the norm of the vector field(Ar/J - r/JA)Xfor any vector field X in To shall be estim.ated by the Lie derivative LeA with respect to
e. By the properties of the Lie derivative we have
(£eA)X =£e(AX)"': A£eX
=(VeA)X - VAXe+AVxe·
Consequently, by the second equation of (2.1) it is reformed to (3.1) g«£eA)X, Y)
= g«Ve'A)X, Y) - g(r/JA2X, Y)+g(Ar/JAX, Y);
for any vector fields X and Y in To. Interchanging X and Y in this equation, we get
g«£~A)Y,X)
= g«VeA)X, Y) - g(r/JA2y,X)+g{Ar/JA~X),
from which combined with (3.1) it follows that (3.2) g«£eA)X, Y) - g«£eA)Y, X)
= -g«A2r/J - 2Ar/JA+r/JA2)X, Y).
Now, we do not necessarily assume that eis principal. So we put Ae = ae+~U, where U is a uni~ vector field orthogonal toeanda and ~ are smooth functions on M. By the direct calculation ·and by
Real hypersurfaces in terms of Lie derivative 165
using the property of the structure tensor c/J, the square of the norm (Ac/J - c/JA)X is given asfollows;
g((Ac/J - c/JA)X, (Ac/J - c/JA)X)
= g((A2c/J - 2Ac/JA +c/JA2)X, c/JX) _/32g(X,U)2, from which together with (3.2) it follows that
(3.3)
g((Ac/J - c/JA)X, (Ac/J - c/JA)X)
= g((£eA)c/JX,X) - g((£eA)X, c/JX) _/32g(X,U)2 for any vector field X in To.
Proof of Theorem 1. By the assumption g((.ceA)X, Y) =0 for any
X and Y in To and (3.3) it turns out to be
(3.4) (Ac/J - c/JA)X =0, f3g(X, U) =0, XETo .
By the above second equation it satisfies /3 = 0, which means that ~
is principal, i.e., A~ = a~. By this property and the first equation of (3.4) the shape operator A must commute the structure tensor c/J.
Namely we have
Ac/J - c/JA= O.
By a theorem due to Okumura [11] in PnC and Montiel and Romero [10] in HnC, this completes the proof.
REMARK 1. By a theorem due to Ki,Kim and Lee [4] the following conditions are equivalent with each other:
(l).ceg =0, (2).c ec/J =0, (3).ceA =o.
According to Theorem 1, it is shown that the restriction of the condi- tion (3) to the distribution To implies (1) and (2). However, we note that ruled real hypersurfaces satisfy
for any vector fields X and Y in To, but it is not of type A.
REMARK 2. Inthe forthcoming paper ([61) the fact thatifit satisfies (£el»X =°for any X inTo, then £ep = °is observed.
4. Ruled real hypersurfaces
This section is concerned with necessary properties about ruled real hypersurfaces. First of all, we recall a ruled real hypersurface M of Mn(c), c=lO. Let 'Y:l~Mn(c)be any regular curve. For any t(El) let M~~l(C)be a totally geodesic complex hypersurface through the point 'Y(t) ofMn(c) which is orthogonal to a holomorphic plane spanned by 'Y'(t) and J'Y'(t). Set M = {xEM~~l(C) :tEll. Then the construction of M asserts that M is a real hypersurface of Mn(c),which is called a ruled real hypersurface. This means that there are many ruled real hypersurfaces ofMn (c). Moreover from this construction we know that the distributionTo defined by To(x) ={XETxM: Xl.ex} for xEM is integrable and its integral manifold is a totally geodesic submanifold Mn-I(C) ofMn(c), c=l0.
Now let us give some fundamental properties of the ruled real hy- persurface M ofMn(c),c=lO. Let us put Ae= ae+pU, where Uis a unit vector orthogonal to eand a and P(P=lO) are sII;looth functions on M. As is seen in [1] and [8], the shape operator A satisfies
(4.1) AU = pe.
Infact,ifwe let AU = pe+'YU+e5W for certain vector fieldW orthog- onal toeand U, then
'Y =g(AU,U) =g(-DuC,U) =g(C, DuU) =g(C, VuU) =O,and
8=g(AU, W) =gC-DuG, W) =g(G,DuW)= g(G, VuW) = 0,
because the distribution To is integrable and its integral manifold is totally geodesic in Mn(c), c=l0,where D and V denotes the Riemann- iaIi cotlhectioIlofMn (c) and M 'respectively. Moreover, f!'om{4.1} it follows that
(4.2) AX=O,
Real hypersurfaces in terms of Lie derivative 167
for any vector field X orthogonal to eand U, because
g(AX,e) = g(Ae,X) = g(o:e+(jU,X) = 0, g(AX, U) =g(AU, X) =(jg(e, X) = 0,and
g(AX,Y) =g(-DxC,Y) = g(C, DxY) = g(C, VxY) =°
for any X and Y in To orthogonal to eand U. Thus from (4.1) and (4.2) it turns out to be
(4.3) Ac/JX = -(jg(X,c/JU)e, c/JAX = 0, XETo, which implies that
(4.4) g«Ac/J - c/JA)X, Y) = O,X, YETo.
Next the covanant derivative V'eA with respect to eis explicitly ex- pressed. Since it satisfies
g«V'eA)X,Y) = g(V'e(AX) - AVeX,Y)
= g(Ve(AX), Y) - g(V'eX, AY), X,VETo, we get, by the direct calculation of the left hand side of the above relation and using the propertyV'ee = c/JAe = (jc/JU by (2.1),
I
0, X = Y =Ujg«V'eA)X, Y) = (j2 g(y,c/JU), X = U, Y..LUj (j2 g(X, c/JU), X .lU, Y = Uj
0, X,Y..LU.
On the other hand, we have
!
0, X = Y=U;g(t/>A2X, Y) = (j2 g(y, c/JU), X = U, Y..LUj
0, X..LU,Y = Uj
0, X,Y..LU.
Because ofc/JAX = °for any X inTo, we get by (3.1) (4.5) g«£eA)X, Y) = (j2 g(X, c/JU)g(Y, U), X, YETo.
This shows that the ruled real hypersurface M of Mn (c) satisfies the condition (1.3).
5. Proof of Theorem 2.
Inthis section we shall give a characterization for ruled real hyper- surfaces. Let M be the real hyper$llrface of Mn(c), c¥O,and assume that the structure vector. is not principal. We put A~ -: a~ +f3U,
where U is a unit vectorhi the holomorphic distributionTo. Then by the assumption the function f3 does not vanish identically onM.
Concerning the ruled real hypersurfaces the following· is proved by the present authors [6].
THEOREM C. LetM bearealhypersurface ofMn (c),c#:O andn?:.3.
Ifit satisfies (5.1)
(5.2)
£eg(X, Y}=0,
g«£eAt/J)X, Y)= 0
for anyvector fields X and Y in the distrib1.ztion To, andifthe structure vector fieldisnot principal and da(~)#:O, ~hen M is locally. congruent to aruled real hypersurface.
Now, let Mo beanqpen subset ofM con~istingof pointsx at which f3(x )#:0. By th~assumption that eis not ppncipal, the set Mo is not empty. Inorder to proye the theorem,it su.:ffitces to show that the equa- tions (5.1) and (5.2)in Theorem C hold for any vector fields X andY inTo on Mo.
We consider first (3.3). By the assumptioi?
g«£eA)X,Y) = f32g(X, ifJU)g(Y,U), We get
g((At/J - t/JA)X, (At/J - t/JA)X) = f32~(X, t/JU)2.
From this equation we caD. calculate the normo~,(At/J - t/JA)X+f3g(X,
~lJ){?~dwec~eas~I~,«?~~~(A<P:-+A)~±~g(X'~l:T)~=O,~ETo.
This me8JlS that (5.1) holds on Mo. . Itis also seen that (5.1) is equivalent to
g«£et/J)X, Y) = 0, X,VETo.
Real hypersurfaces in tenns of Lie derivative 169
Using the property of the Lie derivative and the above equation we can prove that (5.2) holds onMo. In fact, by the direct calculation, we get
9((£EA</> )X, Y)
= 9t(£EA)</>X, Y)+g(A(£E</»X, Y)
= j32 g(</>X, </>U)g(Y, U)+9((£E</»X, AY)
= j32 g(</>X, </>U)g(Y, U) +9((£E</»X, (AY)o)+g(AY,~)9((£E</»X,0
= 132g( fj)X, </>U)g(Y, U) +j3g(Y, U)g(('cE</>)X,0,
where(AY)o denotes the To-component ofAY. Because of9(('cE</»X,0
=-j3g(X, U), we can prove (5.2).
This completes the proof of Theorem 2.
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Department of Mathematics Kyungpook University Taegu 702-701, KOREA
