J. Korean Math. Soc. 33 (1996), No. 1, pp. 137-144
ON THE VOLUME FOR TOTALLY REAL SUBMANIFOLDS OF A KAEHLER MANIFOLD
TAE Ho KANG AND JIN SUK PAK
1. Introduction
One of the fundamental problems in Rlemannian geometry is "How the geometric invariants of Rlemannian manifolds are influenced by the curvature restriction ?".
In this note we shall study the volume of a totally real submanifold P of a Kaehler manifold M and obtain an inequality for the relative volume Vol(M)fVol(P).
Now we recall the definition of n-mean curvature. Let (M,<, >
) be a Riemannian manifold of dimension m with metric < , >.
Let X},··· ,Xn ,Y be orthonormal vectors on M and IT the n-plane spanned by Xl, ... ,Xn . The n-mean curvature ofY and IT is defined by
n
K(Y,IT) :=
L
<R(Y, Xi)Xi, Y >,i=l
where we adopt the following definition for the curvature and the Rle- mannian Christoffel tensor on Mj
R(X,Y)Z = [Vx ,Vy]Z - V[X,Y]Z, R(X, Y, Z, W) =< R(X, Y)Z, W > .
A submanifold P of an almost complex manifold (M,J) is called totally real provided that the almost complex structure J ofM maps
Received December 3,1994.
1991 AMS Subject Classification: 53C25.
Key words: Kaehler manifold, totally real submanifold, volume.
The Present Studies were partially supported by TGRC-KOSEF, and the Ba- sic Science Research Institute Program, Ministry of Education, 1995, Project No.
BSRI-95-1404.
tangent vectors to P into normal vectors. Let P be an n-dimensional totally real submanifold of a Kaehler manifold(M, J)of real dimension 2n. Let N be a unit vector field normal to P defined on an open U of P. Let L be the Weingarten map ofP associated to N. We define the IN-normal curvature ofPat p EU, kJN, as the normal curvature of Pat p in the direction IN Le., kJN(p) =< LJN, IN > (p). We define the JN-mean curvature ofP at p by
1 nH - kJN
HJN(p) = --(trL - kJN)(p) = (p),
n-1 n-1
where H(p) is the mean curvature ofP at p.
We denote by Cpn(..\) the complex projective space of real dimen- sion 2n and holomorphic sectional curvature 4..\, and Rpn(..\) the real projective space with constant sectional curvature ..\. It is well known that there is a natural embedding of real projective space Rpn(..\) as totally real, totally geodesic submanifold ofCpn(..\).
The main result we shall prove is
THEOREM. Let M be a connected compact Kaehler manifold of real dimension 2n with almost complex structure J and P a connected compact totally real submanifold ofM with dimension n. Suppose that, for every pEP and every NE SNpP
and
Then
(1.1 ) Vol(M) < Vol(cpnp.)) Vol(P) - Vol(Rpn(..\)) '
where the holomorphic sectional curvature KH of a 2n-dimensional Kaehler manifold (M,< , >,J) is the restriction of the sectional cur- vature ofM to the holomorphic planes, and SNpP denote the fibre of the unit nonnal bundle SNP at pEP.
On the volume for totally real submanifolds 2. Preliminaries
139
Let M be a Riemannian manifold of dimension m, P a submanifold ofM of dimension n« m). We shall denote by N P(SN P) the (unit) normal bundle of P in M, and NpP(SNpP) the fibre ofNP(SNP) at pEP. Let pEP and N E SNpP. Let 'YN(t) be geodesic of M satisfying'YN(O) = p, 'Y~(O) = N. Let us denote byMl- the orthogonal complement of'Y~(t)in T"'YN(t)M. For Y E T"fN(t)M, setting
R(t)Y :=R(,~(t),Y) 'Y~(t), then R(t) is a self-adjoint map ofMl-.
From now on, M will denote a connected compact Kaehler manifold of real dimension 2n with almost complex structureJ. P will denote a connected compact, totally real submanifold ofM with dimension n.
Let pEP, N E SNpP. Let ef(p,N) := in! {t > 0 I'YN(t) is a focal point ofP}. For everyt E (0,ef(p, N)), there is a neighborhood U of'YN(t) and a neighborhoodV ofpEPsuch that pet) = un{m E
M Id(m,V) = t} is a real hypersurface ofM.
Let S(t) be the Weingarten map of the hypersurfaceP(t) associated to a unit normal vector field N(t) defined on pet) as an extension of
'Y~(t). Let w be the Riemannian volume form ofM, dp that ofP and dN that of the unit sphere sn-l(1). We can define a smooth function (}N(p,t) on ((p,N,t) ESNP x RiO < t < ef(p,N)} by
WC/Net)) = (}N(X, t)tn-1dN1\dP1\dt
Then the fundamental equations ([Gr1, 2]) for the real hypersurface P(t) ofM are given by
(2.1)
(2.2) (2.3)
stet) = S(t)2 + R(t), where stet) := V'-YN(t)S(t), (}~(p, t) __ [ S() n -
1]
(} ( ) - tr t + ,
N X,t t
lim(}N(P,t) = l.
t-O
3. The estimates for the volume element
For pEP, N E SNpP, let {eill$i$n be an orthonormal basis of TpP such that en = IN, and let {Ei(t)}l$i$n be parallel vector fields along ,N(t) such that Ei(O) = ei. Then En(t) = J,N(t). We shall consider the following orthogonal direct decomposition of
where Ht := Span {E1(t),··· , E n-1(t)}.
LEMMA 1. Suppose that, for everyPEP, NE SNpP, KHC/N(t))
~ 4'\, K('N(t),Hd ~ (n-1)'\ andKC/N(t),vt) ~ (n -1)'\. Then (3.1)
where PN('\,P, t) =[cos(JXt) - HJ;f.P) sine JXt)]n-l[cos(2JXt)
The equality in (3.1) is attained if and only if < Lei' ei >=<
Lej, ej >=:13, 1 ::; i,j ::;n-1, and with respect to {Ei h$i$2n-l, Set) and R(t) have the matrix form
(3.2)
8(t)=
A' A,P,t A( A,P,t
n-1
o
o
A'(A,P,t) A( A,P,t)
-..;xcot(..;xt)
n-1
-..;xcot(..;xt)
On the volume for totally real submanifolds
n-1
o
141
R(t) =
o
4A
n-1
where A(A,,8,t)= [cos(JXt) - (-~)sin(..;xt)],
B(A,kJN,t) = [cos(2JXt) - ( kJ:;. )sin(2v1t)].
2VAt and I denotes the derivative with respect to t.
Proof. We adopt an elementary proof due to [G] which is essentially the original proof of [Gr 2]. Let {eI,··· , en-I, en = IN, en+l = Jel, . .. ,e2n-l = Jen-I} be an orthonormal basis of N1. such that the basis {edl~i~nofTpP diagonalizes the Weingarten map LN ofP associated to N, whereN1.is the orthogonal complement of the vector space spanned byN inTpP. And let {Ei(t)h<i<2n-l be parallel vector fields along 'YN(t) such that Ei(O) = ei. - -
Let us consider the functions
(3.1) h(t) =< S(t)Ei(t), Ei(t) > .
Taking the derivative of both sides (3.1), using (2.1) and the Cauchy- Schwarz inequality, we get
(3.2) Ii' ~ li2+< R(t)Ei,Ei >.
Then the hypotheses KbN(t),Ht) ~ (n - l)A, KHbN(t» ~ 4A and K(')'N(t),v,) ~ (n - l)A imply
1 n-l I 1 n-l 1 n-l 2
(n - 1
L
h) ~ (n _1L
Jl) + A ~ (n _1L
h) +A,i=1 i=1 i=1
(3.3) I~ ~ I~ +4A,
1 2n-l I 1 2n-1 2
(n - 1 L h) > (n _1 L h) +A,
i=n+l i=n+1
with J;(O) =< Lei' ei >, i = 1"" ,n, respectively. Then [Gr1] and the first and second inequalities gives
(3.4)
(3.5)
_1-I:r(t) > VAsin(VAt)+HJNcos(VIt) n-1 i=1 I - cos(VIt) - (HJN/VI)sin(VAt)
f > 2VA sin(2VAt)+kJN cos(2VIt)
n - cos(2VIt) - (kJN/2VA)sin(2VIt)
respectively and the denominators of the right hand sides of these in- equalities are positive from t = 0 to the first zero of each one. For i = n+1, ... ,2n - 1 we have J;(O) = -00. Then the third inequality and [Gr1] gives
(3.6)
1 2n-1
- " J;n.- 1 L....J i=n+1
> -VA
tan(VIt) Then, from (2.2)
d n-1 2~1 n-1
dt In8N(p,t) = - [trS(t)+-t-] = - L....J J;(t) - - t - i=l
s:; ~In [(cos(v>:t) - (~)n-l)(cos(2v'Xt) _ kJN sin(2v'Xt»(sineVAt) )n-1]
2VA t '
from which (3.2) follows, taking account of (2.3).
If we have the equality in (3.1), then all inequalities in this proof must be equalities. The first equality in (3.3) implies J;(t) =/jet), 1s:;
i,j s:; n-1, and < Lei' ei >=< Lej, ej >= (3. Equalities in (3.1), (3.4) - (3.6) imply that Ei(t) are eigenvectors ofSet) with eigenvalues
VIsineVAt) +(3 cos(VAt) £
or 1~iS:;n-1
cos(VAt) - ((3/VA) sineVIt) (3.7) J;(t) = 2VAsin(2VAt)+kJNcos(2VAt)
for z=n cos(2VAt) - (k JN /2VA)sin(2VIt)
-VIcot(VAt) for n+1 s:;i s:; 2n - 1
On the volume for totally real submanifolds 143
The matrix form for R(t) follows from (3.7) and the equality in (3.2).
Q.E.D.
REMARK 2. Let ec(p, N) := sup{t > 0 Id(p, "{N(t» = t}. Then ec(p, N) ~ ej(p, N)(see, e.g., [He]). ITM =cpn(A)andP =Rpn(A),
then HJN = 0, kJN = 0, (IN(p,t) = pet) := [Sin;~t)]n-lcos(2v'Xt) and ec(p, N) = 4~ for any pE Rpn(A) and N ESNpP.
Given a function q : X x R+ -+ R, where X is a given space, we denote by z(q) the function which to every x E X associates the first zero of t.he function t -+ q(x,t).
LEMMA 3. Suppose that HJN ~ 0, kJN ~ 0 on P. Then, for each fixed pEP and NE SNpP,
PN(A,p, t) ~ pet), 0 ~ t ~ Z(PN(A,p, t»,
1r
Z(PN(A,p, t» ~ 4v'X
Proof. Letget) = PN~~;f,t). Theng'(t) ~ 0 for 0 ~ t ~ Z(JLN(A,p, t»
and g( 0) = 1. The second assertion follows from the assumption.
Q.E.D.
4. Proof of Theorem
It is well known ([Gr 1, 2]) that Z((JN(p,t» = ej(p,N). Then we have from Lemma 1, 3 and Remark 2
1
ee(p,N)11
Vol(M) = (IN(p, t)tn-1 dN dP dt
o p sn-l(l)
~ 1~o Jp
({
J sn-l(l)p(t)tn-1dN dp dt= Vol(P)
147\
f f tn - 1pet) dN dp dt Vol(Rpn(A» 0 JRpn(>\) Jsn-l(l)Vol(P) n
Vol(Rpn(A» Vol(CP (A». Q.E.D.
REMARK 4. (i) If the equality in (1.1) holds, then P is a totally geodesic submanifold ofM. (ii) IfM = cpnp,), then the equality in (1.1) holds if and only if P = Rpnp,) (from [Ki]). If P = Rpnp,), then the equality in (1.1) holds if and only ifM = cpn(>,) (from [Na]).
References
[C-V] B. Y. Chen and L. Vanhecke, Differential geometry of geodesic spheres, J.
Reine Angew. Math. 325 (1981),28-67.
[Gr1] A. Gray, Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula, Top. 21 (1982), 201-228.
[Gr2] A. Gray, Tubes, Addison-Wesley Publishing Co. (1990).
[G] F. Gimenez, Comparison theorems for the volume of a complex submanifold of a Kaehler manifold,Israel J. of Math. 71 no. 2 (1990),239-255.
[G-M] F. Gimenez and V. Miquel, Volume estimates for real hypersurfaces of a K aehler manifold with strictly positive holomorphic sectional and antihlol- morphic Ricci curvatures, Pacific J. Math. 142 (1990), 23-39.
[He] R. Hermann, Focal points of closed submanifolds of Riemannian spaces, Ned- erI. Akad. Wetensch. Proc. Ser. A66 (1963), 613-628.
[Ki] M. Kimura, Real hypersurfaces and complex submanifolds in complex pro- jective space, Trans. A. M. S. 296 (1986), 137-149.
[Na] S. Nayatani, On the volume of positively curve~Kaehler manifolds, Osaka J. Math. 25 (1988), 223-231.
Tae Ho Kang
Department of Mathematics University of Ulsan
Ulsan 680-749, Korea Jin Suk Pak
Department of Mathematics Kyungpook National University Taegu 702-701, Korea
