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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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manifolds, preprint, 1996.
(2] P. BUEKEN, Reflections and rotations in contact geometry, doctoral dissertation,
Katholieke Universiteit Leuven, 1992.
(3] B. Y. CHEN and L. VANHECKE, Isometric, holomorphic and symplectic reflections,
Geom. Dedicata 29 (1989), 259-277.
(4] B. Y. CHEN and L. VANHECKE, Symplectic reflections and complex space forms, Ann.
Global Anal. Geom. 9 (1991), 205-210.
(5] C. T. J. DODSON, L. VANHECKE and M. E. VÁZQUEZ-ABAL, Harmonic geodesic symmetries,
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[9] O. KOWALSKI, F. PRÜFER a.nd L. VANHECKE, D'Atri spaces, Tapies in Geometry: In
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¡.
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Recibido: 20 Septiembre 1996
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