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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
On the Variety of Planar Normal Sections
Alicia N. García, Walter N. Dal Lago and Cristián U. Sánchez
Partially supported by SECYT-UNC and CONICET, Argentina
Abstract: In the present paper we present a survey of results concerning the variety of planar normal sections associated to a natural embedding of a real flag manifold
. The results included are those that, we feel, better describe the nature of this algebraic variety of
. In particular we present results concerning its Euler characteristic showing that it depends only on
and not on the nature of
itself. Furthermore, when
is the manifold of complete flags of a compact simple Lie group, we present what is, in some sense, its dimension and a large class of submanifolds of
contained in
.
In Differential Geometry, the study of submanifolds is frequently associated to the to the theme of "normal sections". This was already present in works of Euler (1707-1783) when he studied surfaces embedded in . Given a surface
embedded in
we can obtain geometric information about the surface itself and the way it is contained in
via properties of the curves that are obtained by cutting the surface with planes determined by unit tangent vectors and the normal vector to the surface at each point. This curves are called normal sections and they give information about the intrinsic and extrinsic geometry of the surface
Approximately in 1980, Bang Yen Chen generalized this notion for submanifolds of of codimension larger than 1, in the following natural manner:
Let be an isometric immersion and
a point in
We identify a neighborhood of
with its image by
and consider, in the tangent space
a unit vector
If
denotes the normal space to
at
, we may define an affine subspace of
by
If is a small enough neighborhood of
in
then the intersection
can be considered the image of a
regular curve
parametrized by arc-length, such that
. This curve is called a normal section of
at p in the direction of
In a strict sense, we ought to speak of the "germ" of a normal section at
determined by the unit vector
A change in the neighborhood
will change the curve; however, this new curve will coincide with
in a neighborhood of zero. Since our computations with the curve
are done at the point
we may take any one of these curves. We may also assume that
is an embedding.
Since 1980, several authors, for instance Chen, Verheyen, Deprez ([2], [3], [8]) have studied geometric properties de submanifolds of Euclidean spaces in term of their normal sections. They obtained interesting results which characterize submanifolds of where: the geodesic are planar; the normal sections are geodesics; the normal sections have the same constant curvature; etc.
Following B.Y. Chen, we say that the normal section of
at
in the direction of
is pointwise planar at
if its first three derivatives
and
are linearly dependent, i.e. if
In 1982, Chen obtained the following interesting result.
Theorem 1. [2] A spheric submanifold of (i.e. contained in a sphere) has all its normal sections pointwise planar if and only if the second fundamental form is parallel.
This fact was for us of the particular interest because, in 1980, Ferus had related symmetric R-spaces with properties of the second fundamental form through the following:
Theorem 2. [9] A spheric submanifold of is a symmetric R-space if and only if the second fundamental form is parallel.
The previous theorems clearly yield the following:
Theorem 3. A spheric submanifold of is a symmetric R-spaces if and only if all its normal sections are pointwise planar.
2. The variety of planar normal sections
As consequence of the Theorem 3, for a spheric submanifold of
, the set of tangent vectors which define pointwise planar normal sections contains information about whether a given R-space is or not symmetric. So, if for each
, we denote by
the set of
such that
and define pointwise planar normal sections, we have that
is a symmetric R-space if and only if
Therefore, if
is a R-space which is not symmetric we have that
Our first objective was to obtain information about when
is a R-space (also called real flag manifold).
We recall that an R-space or a real flag manifold is an orbit of an s-representation. The reader is referred to [4, p. 225] and references therein, for basic information concerning R-spaces, canonical connections, etc.
For our study we need methods and techniques different from the known ones. The first result that we obtained in this direction was the following.
Theorem 4. [4, (2.5)] If is a natural embedding of a real flag manifold and
is a point in
then the normal section
with
and
is pointwise planar at
if and only if the unit tangent vector
at
satisfies the equation
where is the second fundamental form of the embedding
and
denotes the difference tensor between the Riemannian connection
(associated to the metric induced from the Euclidean metric) and the canonical connection
(associated to the "usual" reductive decomposition of the Lie algebra of the compact Lie group defining
Then, given a point in the real flag manifold
Since clearly implies
we may take
as the image of this set in the real projective space
Since
is an orbit of a group of isometries of the ambient space
it is clear that
does not depend on the point
and we may denote it by
The last theorem allowed us to describe tangent vectors which define pointwise planar normal sections as solutions of an equation and furthermore to obtain the following interesting consequence.
Corollary 1. [4, (2.9)] is a real algebraic variety of
and its natural complexification
is a complex algebraic variety of
, defined both by homogeneous polynomials of degree 3.
These varieties measure, in some sense, how far is the real flag manifold from being a symmetric space (i.e. a symmetric real flag manifold). By Theorem 3,
and
differ respectively from
and
. However, surprisingly enough, they have the same Euler characteristic as we see in the following:
Theorem 5. [4], [14] Let be a real flag manifold and let
be its natural imbedding. Let
be the variety of directions of pointwise planar normal sections at a point
and let
be the natural complexification of
. If
denotes the Euler characteristic with respect to rational coefficients, then
In [4] we gave a proof of this fact when is a complex flag manifold. The methods used in that paper were not strong enough to tackle the general case. However, several years later we were able to obtain the proof for the general case (see [14]).
3. Submanifolds in the variety of planar normal sections
Looking for information about the "size" of we studied the existence of a great deal of smooth subvarieties embedded into
and contained in
when
is a manifold of complete flags of a compact simple Lie group.
In order to indicate our results we need introduce the following notation.
Let be a simply connected, complex, simple Lie group and let
be its Lie algebra. Let
be a Cartan subalgebra of
and
the root system of
relative to
We may write
, where
indicates the set of positive roots with respect to some order.
Let us consider in the Borel subalgebra
Let
be the analytic subgroup of
corresponding to the subalgebra
is closed and its own normalizer in
. The quotient space
is a complex homogeneous space called the manifold of complete flags of
.
Let be a system of simple roots. We may take in
a Weyl basis [12, III, 5]
and
The following set of vectors provides a basis of a compact real form
of
.
![]() | (1) |
We shall denote by the real vector space generated by
and by
that of
Then we may write
.
Let be the analytic subgroup of
corresponding to
.
is compact and acts transitively on
which can be written as
where the subgroup
is a maximal torus in
The manifold
is then a compact simply connected complex manifold. This is the manifold of complete flags for the given compact connected simple Lie group
In the rest of this paper we shall restrict our attention to this case.
It is well known that is the orbit of a regular element
by the adjoint action of
on
. Then we have a natural embedding
of
on
which we may assume isometric by taking in
the inner product given by the opposite of the Killing form.
Then the tangent and normal space to at
are
and
If then
and for the second fundamental form of the embedding
we may write
![]() | (2) |
The coefficients are homogeneous polynomials of degree
in the variables
Then, by Theorem 4,
defines a pointwise planar normal section if and only if
for
3.1. Fat Submanifold. We obtained explicit enough expressions for the polynomials defined by (2), which allowed us to prove that they are
-linearly dependent but this is not the case for any subset of them with
elements.
Theorem 6. [7] The polynomials (
) defined in (2) satisfy:
(i)
(ii) For any such that
the set
is
linearly independent.
With this fact, we can get certain information about the size of
Theorem 7. [7] There is an open set in the variety which is an embedded submanifold in
of dimension
where
and
.
(The topology of is the induced one from the usual topology of
To get this result it was necessary to find points in
, the unit sphere of
, such that they are regular points of the function
whose coordinates are the polynomials
and that satisfy
3.2. Projective subspaces in . Another way to get information about the "size" of
is to know a sufficient amount of projective subspaces in it.
We shall denote by the real projective space associated to a real vector space
Associated to the simple group defining the complex flag manifold
, we have its family of symmetric spaces of type I [12, p. 518] and among them, we want to consider those which are inner, i.e. the spaces in which the symmetry at each point belongs to the group
Among all compact symmetric spaces, these are the only ones strongly related with the algebraic variety
It is well known that each one of the simple groups gives rise to at least one of these symmetric spaces. They are those of the form
where
is a subgroup of maximal rank in
The ones which are not inner in the list in [12, p. 518] are
and
By conjugating if necessary, we may assume that
contains
Let be the Lie algebra of
and write
where
is the orthogonal complement to
with respect to the Killing form. Then
and
The motivation to consider the tangent space to the inner symmetric space
, in our study of the algebraic variety
arises from the following simple fact which provides the first examples of projective subspaces included in
.
Proposition 1. [5, Prop. 4.1] Let be the tangent space of the inner symmetric space
at
Then
Remark 1. [5, Rem. 4.1] For the subspace mentioned in the last proposition, there exists a root
such that
is of the form
and is defined by
.
For this, it was natural to start by studying those subspaces of the tangent space to at
of the form
and such that
The subspaces mentioned above, are exactly those subspaces of the tangent space
which are
-invariant (see for instance [11]).
The first important result to our objective, was the following characterization, in terms of the structure of the Lie algebra of the simple group This characterization was a very useful tool for the proof of many of the results obtained.
Theorem 8. [5, Th. 4.2] Set with
Then
The tangent spaces of the inner symmetric space associated to play an important part among the subspaces
-invariant of the tangent space
. This can be seen in the following two theorems.
Theorem 9. [5, Th. 4.3] Let be the tangent space of the inner symmetric space
at
Then
is maximal among the projective spaces
contained in
with
of the form
for
This theorem is the best we can hope to get for projective subspaces arising from tangent spaces
at the base point
, of irreducible inner symmetric spaces
. We were able to show that if
the projective spaces generated by those
are not maximal among all the projective spaces contained in
(see [5, section 5]).
The irreducible inner symmetric spaces for which
, are the following
These are those whose tangent spaces, at the basic point, are of the form
where is the set of all short roots. We proved that for the spaces of the families BDI, CII and the single space FII, the tangent space
does not generate a maximal projective space in
However these are the only ones with this property as the following result indicates.
Theorem 10. [5, Th. 4.4] Let be the tangent space of the inner symmetric space
at
Then
is maximal in
if and only if
does not vanish.
Another question arises quite naturally. How large can a subspace -invariant defining projective spaces contained in
be?. Clearly an answer to this question yields information about the "size" of the variety
The tangent spaces of the irreducible symmetric spaces are deeply related to this question and, as we expected, they provide the -invariant subspaces
of larger dimension such that
is contained in the variety of planar normal sections.
The list of irreducible symmetric spaces [12, p. 518] indicates that the irreducible inner symmetric spaces of maximal dimension for given groups are those included in the following table with their respective dimensions. We denote them by
and
Theorem 11. [6, Th. 1.1, Th. 1.2] If
is a subspace
-invariant defining a projective subspace in
, then
(i)
(ii) If
then
is tangent to the symmetric space
at a fixed point of the action of the torus
The existence of projective subspaces in the variety of planar normal sections makes it rather special.
In the previous section we gave information about families of projective subspaces in which have deep relation with the tangent spaces of the inner symmetric spaces associated to the simple group
. This subspaces originate in some
- invariant subspaces of the tangent space of
In [11], related to the study of extrinsic symmetric CR-structures on the manifold of complete flags , it was observed that there is a strong connection between the holomorphic tangent spaces of these structures and those subspaces of the tangent space to
which are
- invariant and also give rise to projective subspaces in
. This particular fact throws new light on the interest of the study of these subspaces in
.
In the previous theorems we characterize, for the manifolds of the form , those subspaces
of the tangent space to
which are
-invariant and define projective subspaces of maximal dimension in
. Making a deeper analysis in this direction, for the manifolds
, we continued studying those subspaces
that are
-invariant and define projective subspaces in
but that, in some "interesting" sense, are of "minimal" dimension. These subspaces are those
, which are of minimal dimension and not properly contained in any other
-invariant subspace, defining projective subspaces in
. For these subspaces we have obtained the following:
Theorem 12. [7] Let and
be embedded in
as the orbit of any regular element
. Let
(
) be a subspace of
which is maximal among the subspaces
-invariant of
defining projective subspaces in
Then
(i)
(ii) If then
is the tangent space to the projective space
at a point
where
is defined in [15, p. 80] and
is an element in
, the Weyl group of the pair
Due to the fact that the converse statement of (ii) above is obviously true, this theorem gives a geometric characterization of the subspaces of
which are
-dimensional, defining projective subspaces in
and maximal among the subspaces
-invariant of
.
Joining Theorems 11 and 12, the subspaces of
which are maximal among the subspaces
-invariant of
defining projective subspaces in
, satisfy
and also, when is one of the two ends of the above inequality, the subspace
is tangent to the inner symmetric space of minimal and maximal dimension associated to the group
.
When the subspace is such that
, if we pose no restriction on
and
we cannot assure that
is tangent to some inner symmetric space of the group
Furthermore, we give examples in [7] to show that we cannot even assure that
is tangent to a homogeneous manifold
with
The obtained results allow us to mention the following consequences which we feel are interesting and that in some sense motivated our interest in having a deeper understanding of the projective subspaces in the variety of planar normal section.
Keeping the notation of [11, Th.8] and calling
we may write :
Corollary 2. [7] Let be maximal among the holomorphic tangent spaces at the base point of
-invariant minimal almost Hermitian extrinsic symmetric CR-structure on
Then
(i)
(ii) If or
then
is the tangent space, at some point, to the projective space
or the symmetric space
respectively.
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[2] Chen, B. Y.: Differential geometry of submanifolds. with planar normal sections, Ann. Mat. Pura Appl. 130 (1982), 59-66. [ Links ]
[3] Chen, B. Y. and Verheyen, P.: Sous-variétés dont les sectiones normales son des géodésiques, C.R. Acad. Sci. Paris Ser A 293 (1981), 611-613. [ Links ]
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[5] Dal Lago, W. , García, A. and Sánchez, C.:Maximal projective subspaces in the variety of planar normal sections of a flag manifold, Geom. Dedicata 75 (1999), 219-233. [ Links ]
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[10] Ferus, D.:Immersionen mit paralleler zweiter Fundamentalform: Beispiele und Nicht-Beispiele, Manuscripta math. 12, (1974), 153-162. [ Links ]
[11] García, A. and Sánchez, C.: On extrinsic symmetric CR-structures on the manifolds of complete flags. Beiträge zur Algebra und Geometrie 45 (2004), 401-414. [ Links ]
[12] Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978. [ Links ]
[13] Humphreys J. E. Introduction to Lie Algebras and Representation Theory, Springer-Verlag Berlin. Heidelberg. New York 1972. [ Links ]
[14] Sánchez, C., García, A. and Dal Lago, W.: Planar normal sections on the natural imbedding of a real flag manifold. Beiträge zur Algebra und Geometrie 41 (2000), 513-530. [ Links ]
[15] Wolf, J. and Gray, A. Homogeneous spaces defined by Lie group automorphisms. I. J. Differential Geometry 2 (1968),77-114. [ Links ]
Alicia N. García
FaMAF-CIEM (UNC-CONICET)
Ciudad Universitaria
5000 Córdoba - Argentina.
agarcia@mate.uncor.edu
Walter N. Dal Lago
FaMAF-CIEM (UNC-CONICET)
Ciudad Universitaria
5000 Córdoba - Argentina.
dallago@mate.uncor.edu
Cristián U. Sánchez
FaMAF-CIEM (UNC-CONICET)
Ciudad Universitaria
5000 Córdoba - Argentina.
csanchez@mate.uncor.edu
Recibido: 24 de octubre de 2005
Aceptado: 3 de octubre de 2006