versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.48 n.1 Bahía Blanca ene./jun. 2007
Cristián U. Sánchez, Ana M. Giunta and José E. Tala
This research was partially supported by CONICET, SECYT-UNCba and Universidad Nacional de San Luis, Argentina.
Abstract. In the present paper we study Riemannian and canonical geodesics in a real flag manifold M, considered as curves in the ambient Euclidean space of the natural embedding of M.
2000 Mathematics Subject Classification. Primary 53C30, 53C22; Secondary 53B15, 53A04.
In this paper we present some results concerning an interesting problem in the geometry of submanifolds of Euclidean spaces. Our note originated in a paper by Ferus and Schirrmacher  where those authors considered the problem of determining all the submanifolds of Euclidean space whose geodesics (considered as curves in the Euclidean space) are W-curves i.e. Frenet curves with constant curvatures along the curve, see next section for definitions. Ferus and Schirrmacher obtained an important result which we will describe.
W. Strübing  had shown earlier that symmetric R-spaces (a particular case of real flag manifolds) have the property that all their geodesics, considered as curves in the Euclidean space, are W-curves. Ferus and Schirrmacher obtained the following characterization of symmetric R-spaces based on the behavior of their geodesics as curves in the Euclidean space.
Theorem 1.1. Let be a closed connected Riemannian manifold and an isometric immersion. Then the following properties are equivalent:
(i) For almost every unit-speed geodesic , the image curve is a generic W-curve (definition in the next section).
(ii) is an extrinsic symmetric submanifold in the sense of .
Here the phrase almost every unit-speed geodesic means that the tangent vectors of these geodesics fill the unit-tangent bundle up to a closed set of measure zero. Recall that by , extrinsic symmetric submanifold is equivalent to naturally embedded symmetric real flag manifold.
This result has an immediate consequence which suggests some interesting questions namely:
- For those real flag manifolds which are not symmetric, the geodesics which are generic W-curves must have their tangent vectors contained in a subset of the unit-tangent bundle whose complement does not have measure zero.
This suggests studying Riemannian geodesics on general real flag manifolds (or other submanifolds of Euclidean spaces) trying to consider the following problems:
(i) Determine some subset of the unit-tangent bundle which contains the tangent vectors to those geodesics which are generic W-curves and
(ii) Determine, if possible, some subset of the unit tangent bundle which contains the tangent vectors to all those geodesics which are W-curves.
The present paper is devoted to these problems and contains some information concerning problem (i) for every compact submanifold of an Euclidean space, in particular, for real flag manifolds (Theorem 6.1). It contains also a result about generic canonical geodesics (Theorem 6.2) and an answer to problem (ii) in the case of isoparametric submanifolds of rank 2 in an Euclidean space, (Corollary 7.2).
In every real flag manifold there exists an affine connection naturally associated to its homogeneous structure which is the so called “canonical connection” (see Subsection 3.1). It has proven to be useful to characterize these submanifolds of Euclidean spaces . It is known, , that all the geodesics of this canonical connection, considered as curves in the Euclidean space, are W-curves. This suggests to start by studying the set of “coincidence” of the two types of geodesics. This is an easy task in terms of the difference tensor of the two connections (see Proposition 4.2). This may lead one to think that this set contains all the unit vectors generating Riemannian geodesics which are W-curves but this may not be the case.
In the present paper we introduce the subset of the unit tangent bundle containing all the tangent vectors to Riemannian geodesics that are generic W-curves. This subset, which we shall denote by , is that whose fibre at each point is the real algebraic variety (definition in Section 5). This variety has been introduced in  and many of its properties studied also in ,  and .
It seems hard to determine which is the subset of the unit tangent bundle of a general real flag manifold that contains the tangent vectors to all Riemannian geodesics which are W-curves. So it seems surprising that if our real flag manifold is a homogeneous isoparametric submanifold of rank 2 in an Euclidean space , then the unit tangent vectors to all Riemannian geodesics that are W-curves are contained in (see Corollary 7.2).
If our real flag manifold is symmetric then coincides with the unit tangent bundle. But if is not symmetric then is a set of measure zero in the unit tangent bundle of and this seems to be a nice identification of the set mentioned in the above consequence of Theorem 1.1. The complement of this set is, in this case, open in the unit tangent bundle of .
The paper is organized as follows. In the next section we have the definition of W-curves. In Section 3 we recall the definition of real flag manifold and include a subsection recalling also the definition of the canonical connection. In Section 4 we study the coincidence of Riemannian and canonical geodesics. Section 5 contains the definition of the varieties and and in Section 6 we include the results concerning generic Riemannian and canonical geodesics. Finally Section 7 contains Proposition 7.1 and Corollary 7.2 which concern the question (ii) indicated above.
The authors express their appreciation to the anonymous referee whose comments lead to a considerable improvement of the present paper.
Following Ferus and Schirrmacher  we shall say that a regular curve is a W-curve of rank if, for all , the derivatives are linearly independent, the derivatives are linearly dependent and if the (therefore well defined) Frenet curvatures are constant (independent of the parameter but depending on the geodesic).
We reproduce now a result from  which we need.
Lemma 2.1. Let be a W-curve of infinite length and parameterized by arc length. If the image is bounded, then the rank of is even, . Furthermore, there are pairs of positive constants (unique up to order) and a set of orthonormal vectors in such that
Also following , we shall say that a W-curve in is generic if the real numbers are independent over the rationals. This is equivalent to say that the closure of the image of in is a torus
Real flag manifolds can be informally defined as follows: Let be an irreducible symmetric space (compact or noncompact) a point in and its tangent space at that point. The corresponding isotropy group at acts on by the derivatives (at ) of its elements and its orbits on are the so called R-spaces or real flag manifolds. By their definition, these spaces are compact homogeneous and have a natural embedding in the Euclidean space . The Riemannian metric that is usually considered on them is the induced one. We shall denote by the inner product in and by the Levi-Civita connection associated to the Euclidean metric in .
In the homogeneous space we also have the “canonical” connection which we describe briefly. To that end we “formalize” the definition of real flag manifold given above. The necessary ingredients to construct the arbitrary real flag manifold and its canonical embedding are the following. Let be a real semisimple Lie algebra without compact factors, a maximal compactly embedded subalgebra of and the Cartan decomposition of relative to Let denote the Killing form of ; then can be considered an Euclidean space with the inner product defined by the restriction of to Let be the group of inner automorphisms of . The Lie algebra of may be identified with . Let be the analytic subgroup of corresponding to ; is compact and acts on as a group of isometries. The real flag manifold is, by definition, the orbit of a non zero vector i.e.
This defines also the natural embedding of the real flag manifold in the Euclidean space We take on the Riemannian metric induced by the embedding. Furthermore we shall assume that the embedding is “substantial” or “full” that is, is not contained in any affine hyperplane of . Let us denote by the isotropy subgroup of . Then, as a homogeneous space, .
In general the group is not connected and we shall denote by the connected component of the identity. Let be the Lie subalgebra corresponding to in Let be the orthogonal complement of with respect to the restriction of to (it is negative definite in ). Then is a reductive decomposition, that is, it satisfies We also have
and is one to one from onto
Theorem 3.1. [9, p. 43]Let be a reductive homogeneous space with a fixed decomposition of the Lie algebra , . Then there exists a one to one correspondence between the set of all the invariant affine connections over and the set of all the bilinear functions which satisfy
for each and
Let be the natural projection and assume that we have an invariant affine connection over . We require the following properties for the connection :
(A1) Let be the one parameter subgroup of generated by . Then is a regular curve such that the family of its tangent vectors is parallel along the curve itself.
(A2) Let us consider the curve in Let ; then the parallel translation of the vector tangent at along this curve coincides with translation of by the one parameter subgroup
If the affine connection has property (A2) then it also satisfies (A1).
Proposition 3.2.  The invariant affine connection defined by the function has property (A2) if and only if for each Then, over the reductive homogeneous space, there exists one and only one affine connection which satisfies (A2). It is the one defined by the connection function which is identically zero on .
This invariant affine connection is called the canonical affine connection of second class over with respect to the fixed decomposition of the Lie algebra , We call it the canonical connection. We denote by the Levi-Civita connection associated to the induced metric on from and by the difference tensor defined by . The canonical connection is metric and satisfies . A modern view of the canonical connection can be found in [1, p.203-205].
It is very difficult to determine the Riemannian geodesics of the real flag manifolds which are not symmetric. However, in these spaces the geodesics of the canonical connection are well known. Their geometric properties as curves in the Euclidean space of the natural embedding, were studied in . From that paper we recall
Then we immediately have the following Proposition which is valid for any real flag manifold.
Proposition 4.2. Let be a canonical embedding of the real flag manifold. If is a unitary vector then if and only if the Riemannian and canonical geodesics passing through defined by the vector coincide for all values of their parameter.
Then all these Riemannian geodesics are W-curves.
Let be an isometric immersion and a point in We may identify a neighborhood of with its image by and consider, in the tangent space a unitary 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 the point 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 because a change in the neighborhood will change the curve. However, this new curve will coincide with in the proximity of . 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. 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.
We say that the submanifold is extrinsically homogeneous [1, p.35] if for any two points there is an isometry of such that and .
Given a point in the submanifold we shall denote, as in ,
Since clearly implies we may take the image of this set in the real projective space
If is extrinsic homogeneous in the ambient space , it is clear that does not ”depend” on the point and we may denote it by . In this case is a real algebraic variety of . Its natural complexification , is a complex algebraic variety of . Clearly real flag manifolds are extrinsically homogeneous in their canonical embeddings.
In general for any submanifold of we may define the subset of the unit tangent bundle mentioned above as:
In  the variety of directions of pointwise planar normal sections, of a natural embedding of a real flag manifold was introduced. We refer the reader to ,  and  for the description of diverse properties of this variety. Here we need essentially the definition (2).
If the submanifold at hand is a naturally embedded real flag manifold then we may also define
which is the set of coincidence of the canonical and Riemannian geodesics. The set
is in fact a real algebraic set intersection of the unit sphere in and the affine algebraic variety in defined by the finite set of quadratic polynomials implicit in the condition . Clearly because, for real flag manifolds, one has the identity
proved in .
Let be an isometric embedding of a compact connected manifold . For each point and each unitary vector , let us consider the -geodesic defined by , . We are going to study the curve in considering as the inclusion. We easily compute
We have now the following general result.
Proof. Since is a generic W-curve we have that is a dense set in the torus given by Lemma 2.1. Then the torus is and since we see that the whole torus is contained in . Now it follows from Lemma 2.1 that is tangent to the torus and hence tangent to at . Then (4) yields that . Since the same thing can be proved for any point in the geodesic we have that as claimed.□
A similar result can be proved for the canonical geodesics namely.
Proof. Again is a dense set in the torus given by Lemma 2.1 and set then
Now, since is a real flag manifold,
This theorem corrects [11, p.300,14].
This Corollary could be rephrased as “in a non-extrinsically symmetric canonically embedded R-space, canonical geodesics are generically non-generic”. This is in fact also true for the Riemannian geodesics of any non extrinsic symmetric compact embedded submanifold of as Theorem 6.1 shows. On the other hand, for extrinsic symmetric submanifolds (symmetric R-spaces) the Theorem of Chen  (see also ) says that coincides with the unit-tangent bundle and, one may think, this is the reason why on these last spaces generic geodesics are generic W-curves.
In this section we assume that is isoparametric submanifold of rank 2 in an Euclidean space . These submanifolds can be considered isoparametric hypersurfaces in the unit sphere . Many of them are extrinsically homogeneous, but there are examples which are known to be non-homogeneous. A reference for the present section is  and references there. Recall that for a regular curve in parametrized by arc-length considered as curve in the ambient Euclidean space the first Frenet curvature is given by . The objective here is to indicate the following
Proof. Due to codimension two, is constant along if and only if is parallel in the normal connection and this is equivalent to
We get then, as a corollary in this particular case, an answer to problem (ii) presented in the Introduction.
Unfortunately we cannot prove that there are on any geodesics such that (besides those tangent to the eigendistributions), neither that, if there is one, it is a W-curve.
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Cristián U. Sánchez
CIEM-Conicet y FAMAF,
Universidad Nacional de Córdoba,
5000, Córdoba, Argentina
Ana M. Giunta
Departamento de Matemática,
Facultad de Ciencias Fis. Mat. y Nat.
Universidad Nacional de San Luis,
San Luis, Argentina
José E. Tala
Departamento de Matemática,
Facultad de Ciencias Fis. Mat. y Nat.
Universidad Nacional de San Luis,
San Luis, Argentina
Recibido: 29 de septiembre de 2005
Aceptado: 28 de marzo de 2007