Anti-holomorphic reflections

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Rev.Acad.Canar.Cienc., VIII (Núm. 1), 5 3 - 63 (1996)
ANTI-HOLOMORPHIC REFLECTIONS
J. C. González-Dávila*
Departamento de Matemática Fundamental
Sección de Geometría y Topología
Universidad de La Laguna
La Laguna, Spain
ABSTRACT
L. Vanhecke
Department of Mathematics
Katholieke Universiteit Leuven
Celestijnenlaan 200 B
3001 Leuven, Belgium
We treat anti-holomorphic and !1-reversing reflections with respect to submanifolds
in an almost Hermitian manifold (M,g,J) and investigate the relation with isometric
reflections when (M, g, J) is a Kii.hler or a locally Hermitian symmetric space.
1991 Mathematics Subject Classification. 53B20, 53B25, 53B35, 53C35, 53C55.
Key words and Phrases. Anti-holomorphic and !1-reversing reflections, Kahler manifolds, Hermitian
symmetric spaces, totally real submanifolds.
l. INTRODUCTION
Local geodesic symmetries (that is, reflections with respect to a point) and local reflec­tions
with respect to a submanifold in a Riemannian or pseudo-Riemannian manifold have
been studied intensively and they play an important role at several places. For manifolds or
submanifolds of sorne particular kind, these local diffeomorphisms have sorne special proper­ties
and these properties ma.y in turn be used to characterize sorne special types of ambient
spaces or subma.nifolds. Isometric reflections are the most well-known examples but also
volume-preserving (up to sign) and harmonic reflections have been considered. Moreover,
when the ambient space is an almost Hermitian manifold, a Kahler manifold or a Hermitian
symmetric space, one has considered holomorphic and symplectic reflections and their rela-tion
with isometric ones. We refer to [3], [4], [5], [9], [10], [11], [12], [14], [16], [17], (18], [19]
for a collection of resulta and for further references. Next, we refer to [2], [6], [13], (15], and
the included reference lists, where reflections with respect to curves have been used to de­fine
r,o-symmetric spaces, Killing-tra.nsversally symmetric spaces and transversally symmetric
•supported by the Consejería de Educación del Gobierno de Canarias.
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liations. Finally, we mention [1], [7] far examples where isometric refiections are used to
instruct interesting new examples of a particular class of Riemannian manifolds (namely,
eakly symmetric spaces).
In this note we continue our research about refiections in the framework of (almost) com­ex
geometry and concentrate on anti-holomorphic and !l-reversing refiections with respect
submanifolds. We derive sorne general results and investigate their mutual relation and
e connection with isometric refiections when the ambient space is Kii.hlerian or locally iso­etric
to a Hermitian symmetric space. As we will see and as is already known, totally real
bmanifolds of maximal dimension play a crucial role in this context.
The method to derive the results uses Jacobi vector fields and power series expansions.
e collect the needed material in Section 2. The main results are derived in Section 3.
2. PRELIMINARIES
We start by recalling sorne basic facts and refer to [8], [19] far more details and references.
Let (M,g) be a smooth, n-dimensional Riemannian manifold and let P be a connected,
relatively compact, topologically embedded submanifold of dimension q. All data are supposed
to be analytic where this is needed. Denote by Tl. P the normal bundle of P and by expp the
exponential map of this bundle, that is, exp p( m, v) = expm v far all m E P and all v E r;; P.
The set '.T p( s) defined by
'.Tp(s) = { expp(m, v) J v E Tl. P, , llvJJ < s, m E P}
where s is supposed to be smaller than the distance from P to its nearest focal point, is
said to be the tubular neighborhood of radius s around P. Now, the mapping <pp on '.Tp(s) ,
defined by
<pp : p = expp(m, v) >--+ <pp(p) = expp(m,-v)
far all m E P and all v E r;¡p such that IJvlJ < s, is an involutive local diffeomorphism of
M. P belongs to its fixed point set. This <pp is called a (local) reflection with respect to P.
To describe this map analytically we shall use Fermi coordinates which we introduce
now. Let { Ei, ... , En} be a local orthonormal frame field of ( M, g) defined along P in a
neighborhood of m E P such that Ei, . . . , Eq are tangent to P and Eq+l • ... , En are normal
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to P. Next, let (y1 , ... , yq) be a coordinate system in a neighborhood of m in P for which
8
,,.---,(m)=E;(m), i=l, ... ,q.
uy'
Since every point p in '.T p( s) can be expressed in a unique way as
p = expb ( t t.,E.,)
a=q+l
for sorne b E P, we put
n
x'(expb( L t.,E.,)) y'(b), i=l, ... ,q,
a=q+l
xª( expb ( t t.,E.,)) = ta, a= q+ 1, . . . ,n.
a=q+l
Then ( x1, ... , xn) is a coordinate system on '.T p( s ), called a Fermi coordinate system ( relative
to m, (y1, ... , yq) and (Eq+l• ... , En)). With respect to such a Fermi coordinate system, the
reflection 'i' p takes the following (local) form:
Further, there exists a strong relation between the basic vector fields Iza of the Fermi
coordinate system and sorne special Ja.cobi vector fields along geodesics through m in M. To
describe this rela.tion, we choose a. fixed unit normal vector u at m, u ~ Tj¡P C TmM, a.nd
'¡
consider the geodesic ¡(t) = expm(tu). Further, we adapt the fra.me field (Ei. ... , En) such
that En(m) =u= ¡'(O). Finally, we denote by Y., the Ja.cobi vector fields along ¡ satisfying
the following initial conditions:
Y;(O) = E;(m) Y;'(O) = Vu¡f;.,
Ya(O) O Y;(o) Ea(m)
for all i = 1, . .. , q and a = q + 1, ... , n - l. Here, V denotes the Levi Civita connection of
(M,g). Then we ha.ve
8 8
Y;(t) = ,,.---,(¡(t)), Ya(t) = t;;-;;{r(t)).
ux' ux
Next, let (Fi, . .. , Fn) be the frame field along ¡ obtained by parallel translation of
{Ei. ... , En} and define the endomorphism-valued function t >-+ D.,(t) by
Y.,(t) = D.,(t)F.,, a= 1, .. . , n - l.
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Then this function satisfies the Jacobi equation
n: + R•D,.. =O
where t 1-+ R(t) is the endomorphism-valued function on {'y'(t)}.L C T-y(t)M defined by
R(t)x = R"'f'(t)x 1'(t), x E {"'f'(t)}J...
Here, R is the Riemannian curvature tensor taken with the sign convention
Ruv = V¡u, v] - [Vu, Vvl
for all smooth vector fields U, V. Further we put Rxyzw = R(X, Y, Z, W) = g(RxyZ, W).
The initial conditions for D are given by
where T and l. are defined, via the Levi Civita connection V of P, by
VxY+TxY,
T(N)X + l.xN
for ali smooth X, Y tangent to P and all smooth N normal to P, and
g(T( u)E;, E;)(m),
g(l.E;Ea, En)(m).
Tx Y = T(X, Y) is the second fundamental form operator of P and T(N) is the shape
operator of P with respect to N. They are related by g(T(N)X, Y) = -g(T(X, Y), N).
Further, l.xN = ViN where V'.L is the normal connection along P.
Using the initial conditions for D,..(t), one obtains the following useful power series ex­pansions:.
(1) {
D,..(t)F¡
D,..(t)Fa
E;(m) + t(T E¡ - 1 l.E;)(m) - ~(RE;)(m) + O(t3 ),
tEa(m) - t(REa)(m) + O(t4 ),
for i = 1, . . . , q and a = q + 1, ... , n - l.
We finish this section with a criterion for isometric reflections with respect to P.
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Theorem 2.1. (3] Let (M,g) be a Riemannian manifold and P a submanifold. Then
the reflection cpp is an isometry if and only if
(i) P is total/y geodesic;
(ii) (Vik .. . uR)uvu is normal to P,
(Vtk~1• uR)uvu is tangent to P and
for ali normal vectors u, v of P, any ta'!'gent vector x of P and ali k E N.
Then we get at once:
Corollary 2.1. (3] Let (M,g) be a local/y symmetric Riemannian manifold and P a
submanifold. Then the reflection cpp is an isometry if and only if
(i) P is total/y geodesic;
(ii) Ruvu is normal to P for ali u, v E Tl. P.
3. ANTI-HOLOMORPHIC AND 0-REVERSING REFLECTIONS
Now, we turn to the main contents ofthis note. So, let (M,g,J) be an almost Hermitian
manifold and P a submanifold. Then the re:flection cpp is said to be anti-holomorphic (or
J -reversing) if
(2)
and 0-reversing if
(3)
where O denotes the Kli.hler form on (M,g, J) defined by O(X, Y)= g(X, JY) for all vectors
X, Y tangent to M. Further, Pis called a total/y realor anti-invariant submanifold of (M,g, J)
if JTmP C Tj;P for all m E P (20].
First, we prove
Theorem 3.1. Let (M,g,J) be an almost Hermitian manifold, P a submanifold and
suppose that the reflection cpp is anti-holomorphic or 0-reversing. Then P is total/y geodesic
and total/y real with 2 dlm P = dim M.
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Proof. First, let cpp be !1-reversing. Then, for arbitrary tangent X, Y on M along P we have
from (3)
g(cpp.X,Jcpp.Y) = -g(X,JY).
Further, if X, Y are both tangent or both normal to P, we also have
g(cpp.X,Jcpp.Y) = g(X,JY)
and hence, g(X, JY) =O. This implies that Pis totally real and 2 dim P = dim M.
Next, put
Then (3) yields
(4)
for all i = 1, ... ,q, a= q+ 1, ... ,n-1 and p = expm(tu), u E T;!;P, l/ull =l. Using the
formulas from Section 2 we further have
1
-g(Du(t)F¡, J Du(t)Fa),
t
g(Du(t)F¡,Ju).
Using (1) and taking into account that Pis totally real, we obtain
g(E¡,JEa)(m) + t{g(TE¡,J Ea)+ g(E¡,J'Ea) }(m) + O(t2 ),
g(E¡, Ju)(m) + t{g(T E¡, Ju)+ g(E¡, J'u) }(m) + O(t2 )
where T = T(u). Then (4) yields
g(TE¡,JEa)+g(E¡,J'Ea) o,
g(TE¡,Ju)+g(E,,J'u) =O.
So, we have g(T X, J N) = -g(X, J' N) for all vectors X tangent to P and all normal vectors
N. Now, put Y = J N. Then we have
g(T(X,Y),u) = -g(X,J'JY)
and since the right-hand side is skew-symmetric in X, Y, it follows that T = O and hence, P
is totally geodesic.
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Now, we consider the case where cpp is anti-holomorphic. Then (2) implies
cpp.JX = -Jcpp.X = -JX
for all X tangent to P. Hence, J X is normal. Similarly, we obtain JTl. P C T P. So, P is
totally real and 2 dim P = dim M.
To prove that Pis totally geodesic we first note that the components Je of J with respect
to the Fermi coordinates satisfy
So, since
we get
Next, (2) implies
J~ = -n"'7 g7P , a, (3, 1 = 1, ... , n.
g"'n(p) =O , gnn(p) = 1 , a = 1, ... , n - 1,
Jf(cpp(p))
Jt(cpp(p))
Jf(p),
Jt(p)
and using the power series expansiona for g"'P and f2ap, we get
for a= q + 1, ... , n, which yields as before that Pis totally geó.tesic. •
Now, we suppose that (M,g, J) is a Kii.hler manifold (that is, V J = O) and prove
Theorem 3.2. Let P be a totally real submanifold of a Kiihler manifold with 2 dim P =
dim M. lf the reflection cpp is an isometry, then it is anti-holomorphic or equivalently, f2-
reversing.
Proof. Since cpp is an isometry and P belongs to the fixed point set of it, it is a totally
geodesic submanifold. In this case, and using the differential equation for D and its initial
values, we have
l
n~+ 2 (0) = - ¿: c 1k R(l-k>(o)n:(o).
k=O
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Further, this a.nd the conditions (ii) in Theorem 2.1 yield (see (3])
nt21l(o)v is ta.ngent,
nt21+il(o)v is normal,
for V E r;; P, X E T mP. Hence, we get
nt21l(o)x is ta.ngent,
nt21+1l(o)x is normal
a¡(t) + f3¡(t),
aa(t) + f3a(t)
where a¡, °'ª a.re ta.ngent a.nd (3¡, f3a normal a.long P. (Here, we used the identifica.tion of the
spa.ces {1'(t)}.L via. para.lle! tra.nsla.tion.) Moreover, a¡, f3a a.re even functions of t a.nd (3¡, °'ª
a.re odd functions of t. Since (M,g, J) is Ka.hleria.n a.nd P tota.lly real, we ha.ve
a.nd hence,
n;;( 'PP(P))
I
nin('PP(P))
nab('PP(P))
g(a¡(t), J f3;(t)) + g(f3¡(t), Ja;(t)),
g(aa(t), Jf3b(t)) + g(f3a(t), Jab(t)),
g(a¡(t), J f3a(t)) + g(f3¡(t), Jaa(t)) ,
g(a;(t),Ju),
g(aa(t), Ju)
-n;;(p),
nin(p),
-nab(p).
nia( rpp(p))
nan ( 'P p(p))
nia(p),
-nan(P),
This expresses tha.t 'PP is n-reversing. •
Remark 3.1. Using the sa.me technique a.sin the proof of Theorem 3.2 one ca.n a.lso prove
a. corresponding result for holomorphic subma.nifolds: Let P be a holomorphic submanifold
in a Kiihler manifold such that <pp is isometric. Then <pp is holomorphic or equivalently, ·
symplectic. This result extends the similar one obta.ined in [3] for loca.lly symmetric Kahler
ma.nifolds.
To prove our next result, we consider
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Lemma 3.1. Let P be a submanifold of a Kahler manifold (M,g, J) such that the reflec­tion
'PP is anti-holomorphic or n-reversing. Then Ruvu is normal to p for ali u, V E TJ_ P.
Proof. First, let <pp be 0-reversing. Then (3) implies
(5) Oan('PP(P)) = -Oan(P)
where a = q + 1, . .. , n - l. Since P is tota.lly real and 2 dim P = dim M we ha ve
Oan(P) = tg(E., J'u)(m) - ~g(RE., Ju)(m) + O(t3 ).
So, this and (5) imply
(6) RuvuJu =O
for a.11 normal vectors u, v. Now, put u= o:w+f3z in (6) for arbitrary o:,(3 E R and for arbitra.ry
normal vectors w,z. Using the first Bianchi identity and the Kii.hler identity RxyJzJw
Rxyzw, we then get by considering the coefficient of o:2(3:
(7) 3RwvwJ z - RwzvJ w = O.
Interchanging v and z in (7) yields
(8) 3RwzvJ w - RwvwJ z = O
and so, from (7) and (8), we get RwvwJ z = O or equivalently, Rwvw is normal to P along
P.
Fina.lly, if 'PP is anti-holomorphic, a same procedure and J: = nn., J:( <pp(p)) = -J:(p ),
yields the required result. , •
From this Lemma 3.1, Corolla.ry 2.1, Theorem 3.1 and Theorem 3.2 we now derive at once
Theorem 3.3. Let P be a totally real submanifold of a locally Hermitian symmetric
space such that 2 dim P = dim M. Then the following statements are equivalent:
(i) <pp is an isometry;
(ii) <pp is anti-holomorphic;
(iii) <pp is f!-reversing.
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CorolÍary 3.1. Let P be a submanifold of a locally Hermitian symmetric space (M, g, J).
Then the reflection 'f'P is anti-holomorphic if and only if it is !l-reversing.
Finally, using (3, Corollary 4] we have
Corollary 3.2. Let (M,g,J) be a Kahler manifold of constant holomorphic sectional
curvature e # O. Then 'f'P is anti-holomorphic if and only if P is a totally geodesic and totally
real submanifold with 2 dim P = dim M.
Remark 3.2. For more information about the existence of fixed point sets of anti­holomorphic
involutions (that is, real forms) in Hermitian symmetric spaces and for further
references, we refer to (1]. There one also finds references concerning the theory of locally
and globally reflective submanifolds P in M, that is, submanifolds P such that the reflection
'f'P is a well-defined local or global isometry with P as fixed point set.
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Recibido: 20 Septiembre 1996
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