versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
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 (, , ) 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.  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.  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:
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. ,  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  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 ).
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 .
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
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.
(ii) For any such that the set is linearly independent.
With this fact, we can get certain information about the size of
Theorem 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
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 ).
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
(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 , 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.  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
(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 .
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  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.  Let be maximal among the holomorphic tangent spaces at the base point of -invariant minimal almost Hermitian extrinsic symmetric CR-structure on Then
(ii) If or then is the tangent space, at some point, to the projective space or the symmetric space respectively.
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Alicia N. García
5000 Córdoba - Argentina.
Walter N. Dal Lago
5000 Córdoba - Argentina.
Cristián U. Sánchez
5000 Córdoba - Argentina.
Recibido: 24 de octubre de 2005
Aceptado: 3 de octubre de 2006