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## Revista de la Unión Matemática Argentina

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*versión impresa* ISSN 0041-6932*versión On-line* ISSN 1669-9637

### Rev. Unión Mat. Argent. v.48 n.1 Bahía Blanca ene./jun. 2007

**Geodesics and Normal Sections on Real Flag Manifolds**

**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 [8] 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 [13] 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.** [8]*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* [7]*.*

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 [7], 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 [10]. It is known, [11], 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 [4] and many of its properties studied also in [12], [6] and [5].

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 [8] 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 [8] 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 [8], 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

| (1) |

and is one to one from onto

3.1. **The two connections.** We recall the following fundamental facts.

**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.** [9] *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].

**4. Canonical and Riemannian Geodesics.**

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 [11]. From that paper we recall

**Proposition 4.1.** *Let* *be a canonical embedding of the real flag* *manifold* *. For each point* *and each unitary vector* *, if* *is* *the* *-geodesic defined by* *, then* *is a W-curve in* *.*

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 [4],

| (2) |

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 [4] 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 [4], [12] and [5] 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

| (3) |

proved in [4].

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

| (4) |

We have now the following general result.

**Theorem 6.1.** *Let* *be an isometric embedding of a compact connected* *manifold. If* *is a Riemannian geodesic and* *is a generic W-curve* *in* *then, for each* *,*

*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.

**Theorem 6.2.** *Let* *be a canonical embedding of the real flag manifold.* *If* *is a canonical geodesic and* *is a generic W-curve in* *then, for each* *,*

*Proof.* Again is a dense set in the torus given by Lemma 2.1 and set then

| (5) |

Now, since is a real flag manifold,

and because also in this case is tangent to the torus and hence tangent to at , we must have (by (5)) and by (3) □

This theorem corrects [11, p.300,14].

**Corollary 6.3.** *In a non-extrinsically symmetric canonically embedded R-space, the* *vectors of the unit-tangent bundle that generate canonical geodesics which are* generic *W-curves are contained in* *.*

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 [3] (see also [4]) 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.

**7. The case of isoparametric submanifolds of rank two.**

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 [1] 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

**Proposition 7.1.** *Let* *be a compact isoparametric submanifold of rank* 2 *in a* *Euclidean space* *and* *a regular curve parametrized by arc-length in* *.* *Then* *is constant, if and only if,* *for* *each* *.*

*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.

**Corollary 7.2.** *Let* *be a Riemannian geodesic in* *which is a W-curve in* *. Then for each* *we have*

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.

[1] Berndt J., Console S. & Olmos C.: Submanifolds and holonomy. Chapman & Hall/CRC Research notes in Mathematics # 434, 2003. [ Links ]

[2] Bodrenko I.I.: A characteristic condition for an *n*-dimensional sphere in the Euclidean space . *Russian Acad. Sci. Sb. Math. Vol 83, (1995).* [ Links ]

[3] Chen, B. Y.: Differential geometry of submanifolds with planar normal sections, *Ann. Mat.* *Pura Appl.* **130** (1982), 59-66. [ Links ]

[4] Dal Lago, W. , García, A. and Sánchez, C.U.: Planar normal sections on the natural imbedding of a flag manifold, *Geom. Dedicata* **53** (1994), 223-235. [ Links ]

[5] Dal Lago, W. , García, A. and Sánchez, C.U.: Projective subspaces in the variety of normal sections and tangent spaces to a symmetric space. *J. Lie Theory* **8** (1998) 415-428. [ Links ]

[6] 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 ]

[7] Ferus, D.: Symmetric submanifolds of Euclidean space. *Math. Ann*. **247** (1980), 81-93. [ Links ]

[8] Ferus D. & Schirrmacher S.: Submanifolds in Euclidean Space with Simple Geodesics. *Math.* *Ann*. **260** (1982), 57-62. [ Links ]

[9] Nomizu K.: Invariant affine connections on Homogeneous Spaces. American J. Math. **LXXVI**, (1954) 33-65. [ Links ]

[10] Olmos C. & Sánchez C.U.: A geometric characterization of the orbits of s-representations. *J.* *Reine Angew. Math.* **420**, (1991), 195-202. [ Links ]

[11] Sánchez C.U., Dal Lago W. García A. Hulett E.:, On some properties which characterize symmetric and general R-spaces. *Differential Geometry and its Applications* **7** (1997), 291-302. [ Links ]

[12] Sánchez C.U., García A., Dal Lago W.: Planar normal sections on the natural imbedding of a real flag manifold. Beiträge zur Algebra und Geometry (Contributions to Algebra and Geometry) **41**, (2000), #2, 513-530. [ Links ]

[13] Strübing W.: Symmetric submanifolds of Riemannian manifolds. *Mathematische Annalen* **245** (1979), 37-44. [ Links ]

*Cristián U. Sánchez*

CIEM-Conicet y FAMAF,

Universidad Nacional de Córdoba,

5000, Córdoba, Argentina

csanchez@mate.uncor.edu

*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*