Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 2, 513-530.

Planar Normal Sections on the Natural Embedding of a Real Flag Manifold

Cristián U. Sánchez Alicia N. García Walter Dal Lago^{1}

Facultad de Matemática Astronomía y Física Universidad Nacional de Córdoba, Ciudad Universitaria 5000, Córdoba, Argentina e-mail: csanchez@mate.uncor.edu

Abstract. This paper contains a proof of the following result which is an extension of the main result in [5], to the general case of real flag manifolds (also called Rspaces or orbits of s-representations). THEOREM. Let M be a real flag manifold and let j : M → 𝔭 its canonical embedding. Let X [M] ⊂ RP^{n−1}, (n = dim M) be the variety of directions of pointwise planar normal sections at a point of M and let X_{c} [M] ⊂ CP^{n−1} be the natural complexification of X [M]. Let χ denote the Euler-Poincare characteristic then

\begin{matrix}(i) χ (X [M]) = & χ (RP^{n−1}) = & \{ \begin{matrix}0 \\ 1 \end{matrix} & \begin{matrix}n even \\ n odd \end{matrix} \\ (ii) χ (X_{c} [M]) = & χ (CP^{n−1}) = & n. \end{matrix}

MSC 2000: 53C30, 53C42.

1. Introduction

In [5] we introduced the variety X [M] of directions of pointwise planar normal sections and its natural complexification X_{c} [M] , of a natural embedding of a real flag manifold (also called R-spaces or orbits of s-representations). That paper continued the pioneering work of B. Y. Chen [3] and other authors ([4], [6] etc.) on normal sections of submanifolds of Euclidean spaces. X [M] is a real algebraic variety in the real projective space RP^{n−1} and X_{c} [M] is a complex variety in CP^{n−1} where n is the dimension of M. To some extent, they measure the differences between the given manifold M and a symmetric real flag manifold.

1This work has been partially supported by CONICOR (Córdoba, Argentina), CONICET (Argentina)

and SECYT-UNCba (Córdoba, Argentina).

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

The varieties X [M] ⊂ RP^{n−1} and X_{c} [M] ⊂ CP^{n−1} are quite different from their ambient projective spaces however, surprisingly, the main result in [5, p.224, (1.2)] is the following

Theorem 1. Let M^{n} = G/K be a complex flag manifold and let j : M → 𝔤 be one of the natural embeddings of M in the Lie algebra 𝔤 of G. Let X [M] ⊂ RP^{n−1} be the variety of directions of pointwise planar normal sections at a point p ∈ M and X_{c} [M] ⊂ CP^{n−1} the natural complexification of X [M]. If χ denotes the Euler characteristic then

\begin{aligned}(i)  χ (X [M]) & = & χ (RP^{n−1}) & = 0 \\ (ii)  χ (X_{c} [M]) & = & χ (CP^{n−1}) & = n & = dim M. \end{aligned}

The methods of [5] were not sufficiently strong to get this result for the general case of real flag manifolds (R-spaces). The objective of the present paper is to give a proof of this result in the general case.

Our result is the following

Theorem 2. Let M^{n} be a general real flag manifold (also called R-space or orbit of an srepresentation) and let j : M → 𝔭 be its natural imbedding. Let X [M] ⊂ RP^{n−1} the variety of directions of pointwise planar normal sections at a point p ∈ M and X_{c} [M] ⊂ CP^{n−1} the natural complexification of X [M]. If χ denotes the Euler characteristic then

(i)  χ (X [M]) = χ (RP^{n−1}) = \{ \begin{matrix}0 \\ 1 \end{matrix} \begin{matrix}if dim M is even \\ if dim M is odd \end{matrix}
(ii)  χ (X_{c} [M]) = χ (CP^{n−1}) = n = dim M.

As we indicated above this result contains the previous one since every complex flag manifold is an even dimensional real flag manifold.

The paper is naturally divided into two parts considering the real and complex cases which require different arguments.

The next section contains the required notation and basic results. Section 3 contains the required arguments for the real case and Section 4 the proof of the first part of the main theorem. Sections 5 and 6 are devoted to the complex case.

2. Basic facts

In the present section we introduce some of the basic notation to be used throughout the paper. All unexplained notation will have the same meaning as in [5].

Let M be a Riemannian manifold. Let j : M → R^{N} be an isometric immersion and p a point in M. We may identify a neigborhood of p with its image by j and consider, in the tangent space T_{p} (M) , a unitary vector X. If T_{p} (M)^{⊥} denotes the normal space to M at p, we may define an affine subspace of R^{N} by

S (p, X) = p + Span {X, T_{p} (M)^{⊥}} .

If U is a small enough neigborhood of p in M, then the intersection U ∩ S (p, X) can be considered the image of a C^{∞} regular curve γ (s) , parametrized by arc-length, such that γ (0) = p, γ′ (0) = X. This curve is called a normal section of M at p in the direction of X. In a strict sense, we ought to speak of the “germ” of a normal section at p determined by the unit vector X. A change in the neigborhood U will change the curve; however, this new curve will coincide with γ in the proximity of zero. Since our computations with the curve γ are done at the point p, we may take any one of these curves. We may also assume that j is an embedding.

Following B. Y. Chen, we say that the normal section γ of M at p in the direction of X is pointwise planar at p if its first three derivatives γ′ (0) , γ′′ (0) and γ′′′ (0) are linearly dependent, i.e. if γ′ (0) ∧ γ′′ (0) ∧ γ′′′ (0) = 0.

In [5] we studied the pointwise planar normal sections of an orbit of an s-representation (i.e. of the natural embedding of an R-space or real flag manifold). In order to mention one of the results obtained there, we need to recall some strictly necessary notation.

Let j : M → R^{N} be the natural embedding of an R-space and let ∇ denote the Riemannian connection associated to the metric induced from the Euclidean metric ⟨., .⟩. Let ∇^{c} denote the canonical connection associated to the “usual” reductive decomposition of the Lie algebra of the compact Lie group defining M. Let D = ∇ − ∇^{c} denote the difference tensor and α be the second fundamental form of the embedding j. The indicated result is the following.

Theorem 3. [5, (2.5)] If j : M → R^{N} is the natural embedding of an R-space and p is a point in M, then the normal section γ with γ (0) = p and γ′ (0) = X is pointwise planar at p if and only if the unit tangent vector X at p satisfies the equation

α (X, D (X, X)) = 0.

Given a point p in the R-space M we may consider, in the sphere of radius 1 in T_{p} (M) , the subset

X̂_{p} [M] = {X ∈ T_{p} (M) : ∥X∥ = 1, α (X, D (X, X)) = 0} .  (1)

Since X ∈ X̂_{p} [M] clearly implies −X ∈ X̂_{p} [M] , we may take the image X_{p} [M] of this set in the real projective space RP^{n−1}. Since M is an orbit of a group of isometries of the ambient space R^{N}, it is clear that X_{p} [M] does not depend on the point p and we may denote it by X [M]. It is not hard to prove [5, p.227, (2.9)] that X [M] is a real algebraic variety of RP^{n−1} defined by homogeneous polynomials of degree 3. Then X_{c} [M] , the natural complexification of X [M], is a complex algebraic variety of CP^{n−1}.

The necessary ingredients to construct the arbitrary R-space M and its canonical embedding are the following. Let 𝔤 be a real semisimple Lie algebra without compact factors, 𝔨 a compactly imbedded subalgebra of 𝔤 and 𝔤=𝔨 ⊕ 𝔭 the Cartan decomposition of 𝔤 relative to 𝔨. Let B denote the Killing form of 𝔤; then 𝔭 can be considered a Euclidean space with the inner product defined by the restriction of B to 𝔭. Let G = Int(𝔤) be the group of inner automorphisms of 𝔤. The Lie algebra ad(𝔤) of G may be identified with 𝔤. This has the effect of identifying the adjoint action of G on its Lie algebra ad(𝔤) with the natural action of G on 𝔤. Let K be the analytic subgroup of G corresponding to 𝔨 ; K is compact and acts on 𝔭 as a group of isometries. The R-space M is, by definition, the orbit of a nonzero vector E ∈ 𝔭 i.e., M = K.E. This defines also the natural embedding j̇ : M → 𝔭 of the R-space M into the Euclidean space (𝔭, B). We take on M the Riemannian metric induced by this embedding. Furthermore we will assume that the natural embedding is full, i.e. j (M) is not contained in any affine hyperplane of 𝔭.

Let us denote by K_{E} the isotropy subgroup of the point E and then, as a homogeneous space M = K/K_{E}. In general the group K_{E} is not connected and we denote by [K_{E}]_{e} its connected component of the identity. Let 𝔨_{E} be the Lie subalgebra corresponding to [K_{E}]_{e} in 𝔨. Let 𝔪 be the orthogonal complement of 𝔨_{E} with respect to the restriction of B to 𝔨 (it is negative definite on 𝔨). Then 𝔨=𝔨_{E}⊕𝔪 is a reductive decomposition, i.e. Ad(K_{E})𝔪 ⊂ 𝔪. Furthermore we have

T_{E}(M) = [𝔪, E] = [𝔨, E]

and also ad(E) is one to one on 𝔪 and onto T_{E}(M).

One has the following fundamental existence theorem.

Theorem 4. [9, p. 43] Let K/K_{E} be a reductive homogeneous space with a fixed decomposition of the Lie algebra 𝔨 = 𝔨_{E}⊕𝔪, Ad (K_{E}) 𝔪 ⊂ 𝔪 . There exists a one-to-one correspondence between the set of all invariant affine connections ∇^{c} on K/K_{E} and the set of all bilinear functions ω : 𝔪 × 𝔪 → 𝔪 which satisfy

Ad (h) ω (X, Y) = ω (Ad (h) X, Ad (h) Y)

for each X, Y ∈ 𝔪 and h ∈ K_{E}. The correspondence is given by

ω (X, Y) = (∇_{X^{∗}}^{c}Y^{∗})_{E} .

Let π : K ⟶ K/K_{E} denote the natural projection and assume that we have an invariant affine connection ∇^{c} on K/K_{E}. We want that the connection ∇^{c} may have the following properties. (A1) Let exp(tX) be the one parameter subgroup of K generated by X ∈𝔪. Then π(exp(tX))

= (exp(tX)) E is a regular curve such that its family of tangent vectors is parallel along the curve itself.

(A2) Let us consider the curve π (exp(tX)) = (exp(tX)) E in K/K_{E}. Let Y ∈ 𝔪 ; then the parallel displacement of the vector Y, tangent at E, along this curve coincides with the translation of Y by the one parameter subgroup exp(tX).

It is clear that if the affine connection ∇^{c} has the property (A2) then it also satisfies (A1).

Proposition 5. [11, p. 49] The invariant affine connection defined by the function ω satisfies (A2) if and only if ω (X, Y) = 0 for each X, Y ∈ 𝔪. Then on a reductive homogeneous space there exists one and only one affine connection which satisfies (A2). It is defined by the connection function which is identically zero on 𝔪 × 𝔪.

This invariant affine connection is called the canonical affine connection of the second kind on K/K_{E} with respect to a fixed decomposition of the Lie algebra 𝔨 = 𝔨_{E}⊕𝔪.

3. Lemmas for the real case

Let T ⊂ [K_{E}]_{e} be a maximal torus which we shall keep fixed. The torus T acts on 𝔭 and on M by isometries and we may consider in both spaces the respective sets of fixed points, namely F = F (T, 𝔭) and F (T, M). Of course, F is a subspace which contains the vector E ∈ 𝔭 and F (T, M) may have several components (all of them compact, totally geodesic submanifolds of M). Let M_{E} = [F (T, M)]_{E} denote the connected component containing the point E. It is clear that M_{E} = [M ∩ F]_{E}.

At the point E we have the tangent space T_{E} (M) and the subspace V = T_{E} (M_{E}) ⊂ T_{E} (M). It is clear that

V = {x ∈ T_{E}(M) : Ad(g)x = x ∀g ∈ T } ⊂ F

and we can determine a subspace 𝔲 ⊂ 𝔪 such that

V = [𝔲, E] .

The subspace F is, of course, determined by our choice of the torus T ⊂ [K_{E}]_{e}. However the subspace F is a proper subspace of 𝔭 for any torus T, as the next lemma shows.

Lemma 6. For every maximal torus T ⊂ [K_{E}]_{e} , F = F (T, 𝔭) ⫋ 𝔭.

Proof. Let us see first that there exists at least one maximal torus in [K_{E}]_{e} with the required property.

If for every maximal torus T ⊂ [K_{E}]_{e} the subspace F coincides with 𝔭 then the subgroup [K_{E}]_{e} acts trivially on 𝔭. This implies that the group [K_{E}]_{e} which is normal in K_{E} is also normal in K.

Now according to [7, p.252, 1.1,(iii)] we may write every element in G as (exp X) g for some X ∈ 𝔭, g ∈ K and then

(exp X) g [K_{E}]_{e} g^{−1} (exp (−X)) = (exp X) [K_{E}]_{e} (exp (−X)) .
Now for each h ∈ [K_{E}]_{e} we have
(exp X) h (exp (−X)) = (exp X) h (exp (−X)) h^{−1}h =
= (exp X) (exp Ad (h) (−X)) h =
= (exp X) (exp (−X)) h = h,

and this proves that the subalgebra 𝔨_{E} = Lie ([K_{E}]_{e}) is a compact ideal in the semisimple Lie algebra 𝔤 which is supposed to contain no compact factors. This contradiction proves that there is at least one maximal torus T such that F = F (T, 𝔭) ⫋ 𝔭.

Now any other maximal torus in [K_{E}]_{e} is of the form gT g^{−1} for some g ∈ [K_{E}]_{e}. Let F_{g} = F (gT g^{−1}, 𝔭), then F_{g} ⊃ Ad (g) F because, if Y ∈ Ad (g) F then Y = Ad (g) X for some X ∈ F and then for h ∈ T

Ad (ghg^{−1}) Y = Ad (g) X = Y.

If Z ∈ F_{g} then Ad (h) Ad (g^{−1}) Z = Ad (g^{−1}) Z and so Ad (g^{−1}) Z ∈ F which shows that Z ∈ Ad (g) F and proves that F_{g} ⊂ Ad (g) F.

Then F_{g} = Ad (g) F and this shows that if F is a proper subspace of 𝔭 then so is F_{g} and the lemma follows. □

Corollary 7. For any maximal torus T ⊂ [K_{E}]_{e} , [F (T, M)]_{E} ⫋ M.

Proof. By the previous lemma, for any maximal torus T ⊂ [K_{E}]_{e} , F is a proper subspace of 𝔭. If for this torus we have [F (T, M)]_{E} = M then M ⊂ F and this contradicts our assumption that the embedding is full. □

Since ∇^{c} is the canonical affine connection of the second kind in M the ∇^{c}-geodesics through E in M are of the form Ad (exp (tY)) E for Y ∈ 𝔪.

Lemma 8. For X ∈ 𝔲, the ▽^{c}-geodesic in M Ad (exp (tX)) E is contained in M_{E} for every t ∈ R.

Proof. Let g be any element of T. We have

Ad (g) Ad (exp (tX)) E = Ad (g) Ad (exp (tX)) Ad (g^{−1}) E =
= Ad (exp (tAd (g) X)) E = Ad (exp (tX)) E.

This means that Ad (exp (tX)) E ∈ F and hence Ad (exp (tX)) E ∈ [M ∩ F]_{E} = M_{E}. □

Let us denote by C_{K} (T) the centralizer of the torus T ⊂ [K_{E}]_{e} in the group K.

Lemma 9. The group C_{K} (T) acts transitively on M_{E}.

Proof. If X ∈ 𝔲, then the monoparametric subgroup {exp (tX) : t ∈ R} is contained in the group C_{K} (T) because all its elements commute with T. It is well known that the group C_{K} (T) is connected and therefore the orbit Ad (C_{K} (T)) E is connected and it is contained in M_{E}. Furthermore this orbit is clearly closed in M_{E} by compactness.

But it is also open in M_{E} because

T_{E} (Ad (C_{K} (T)) E) ⊇ [𝔲, E] = V = T_{E} (M_{E})

and therefore

T_{E} (Ad (C_{K} (T)) E) = T_{E} (M_{E}) .

This proves the lemma. □

Let 𝔠 ⊂ 𝔨 be the Lie subalgebra corresponding to the subgroup C_{K} (T). Then 𝔲 ⊂ 𝔠. Let now

𝔠_{E} = {X ∈ 𝔠 : [X, E] = 0}

we have the following

Lemma 10. 𝔠 = 𝔠_{E} ⊕ 𝔲 and this decomposition is reductive.

Proof. It is clear from the previous lemma that 𝔠 = 𝔠_{E} + 𝔲 and since ad (E) is injective on 𝔲,this is a direct sum.

According to the definition [9, p. 41] we need to check that Ad (g) 𝔲 ⊂ 𝔲, ∀g ∈ C_{K} (T) such that Ad(g)E = E.

Since

𝔲 = {X ∈ 𝔪 : Ad(h)X = X, ∀h ∈ T } .
Ad(g)𝔲 = {Ad(g)X ∈ 𝔪 : Ad(h)X = X, ∀h ∈ T }

but, for X ∈ 𝔲,

Ad(g)X = Ad(g)Ad(h)X = Ad(h)Ad(g)X ∀h ∈ T

and then

Ad (g) 𝔲 ⊂ 𝔲

and the lemma is proved. □

Let us recall that a submanifold M_{E} of a manifold M with an affine connection ∇^{c} is said to be autoparallel [11, III, p. 32] if ∇^{c}-parallel translation in M along a curve in M_{E} always takes vectors tangent to M_{E} into vectors tangent to M_{E}.

It is possible to “induce” an affine connection into the autoparallel submanifold called the induced connection.

Lemma 11. M_{E} is ∇^{c}-autoparallel in M and the induced connection coincides with the canonical connection in M_{E} associated to the reductive decomposition 𝔠 = 𝔠_{E} ⊕ 𝔲 .

Proof. According to [11, p. 32, 14] M_{E} is autoparallel if and only if for each pair of fields X and Y tangent to M_{E}, ∇_{X}^{c}Y is also tangent to M_{E}.

Let a be a point in M_{E}. Given two fields X and Y tangent to M_{E} we need to prove that ∇_{X}^{c}Y |_{a} ∈ T_{a} (M_{E}).

We know that the connection ∇^{c} is invariant by the action of the group K on M i.e. for each g ∈ K

dAd (g) ∇_{X}^{c}Y = ∇_{dAd(g)X}^{c}dAd(g)Y.

But dAd(g) = Ad(g) and therefore the invariance means

Ad (g) ∇_{X}^{c}Y = ∇_{Ad(g)X}^{c}Ad(g)Y.

Then, for every g ∈ T, we have

Ad (g) ∇_{X}^{c}Y = ∇_{X}^{c}Y

which shows, in particular, that

∇_{X}^{c}Y |_{a} ∈ F.

Since this belongs also to T_{a} (M) we have

∇_{X}^{c}Y |_{a} ∈ F ∩ T_{a} (M) = T_{a} (M_{E}) .

We want to prove now that the induced connection coincides with the canonical connection of the reductive decomposition 𝔠 = 𝔠_{E} ⊕ 𝔲 . Let us denote by ∇^{c1} this last connection.

It is clear now that the induced connection ∇^{c} satisfies the axiom (A2) because the ∇^{c}- geodesics in M generated by vectors in V remain in M_{E} (see Lemma 8) and the ∇^{c}-parallel translation along these geodesics is the same as in M.

Then by Proposition 5 we have

∇^{c1} ≡ ∇^{c}.

This completes the proof of the lemma. □

Let us set the notation L = C_{K} (T) and 𝔩 = Lie (C_{K} (T)). Let Q ⊂ F be the subspace generated by M_{E}. This subspace is clearly invariant by the group L and V ⊂ Q.

Proposition 12. We have in Q a representation ρ of the group L (with induced representation ρ_{∗} : 𝔩 → 𝔤𝔩(Q)) and M_{E} is an orbit of this representation. This defines a full isometric embedding of M_{E} in Q.

The second fundamental form of this embedding is parallel with respect to the canonical connection ∇^{c} on M_{E}.

Proof. Clearly the representation ρ : L → GL(Q) is just the restriction to L of the adjoint representation of K on 𝔭, i.e. ρ(g)U = Ad(g)U ∀g ∈ L, U ∈ Q.

By the definition of Q, the orbit M_{E} = Ad(L)E clearly defines a full isometric embedding of M_{E} into Q.

In order to compute the canonical covariant derivative of the second fundamental form α of M_{E} in Q we resort to [8, p. 359, Lemma 12] which we reproduce, with our present notation, for the benefit of the reader.

Lemma 13. For x, y, z arbitrary tangent vectors in T_{E}(M) the following formulas hold: (i) α_{E} (y, z) =⊥ [ρ_{∗}(Y)ρ_{∗}(Z)E] (ii) D_{E} (y, z) = ⊤ [ρ_{∗}(Y)ρ_{∗}(Z)E] (iii) (▽_{x}^{c}α) (y, z) =⊥ [ρ_{∗}(X)α_{E} (y, z)] where X, Y, Z ∈ 𝔲 are such that x = ρ_{∗}(X)E, y = ρ_{∗}(Y)E and z = ρ_{∗}(Z)E (x, y, z are in T_{E}(M), ⊤ and ⊥ indicate tangent and normal component respectively).

Now we compute the canonical covariant derivative of the second fundamental form α of M_{E} in Q

(▽_{x}^{c}α) (y, z) =⊥ [ρ_{∗}(X)α_{E} (y, z)] ∀ X, Y, Z ∈ 𝔲.

To compute ρ_{∗}(X)α_{E} (y, z) , we use the structure of our R-space M. Since

α_{E} (y, z) =⊥ [ρ_{∗}(Y)ρ_{∗}(Z)E]

where y = ρ_{∗}(Y)E and z = ρ_{∗}(Z)E, if ξ is any normal vector to M_{E} at the point E, we have

B (A_{ξ}y, z) = B (ξ, α_{E} (y, z)) = B (ξ, ⊥ [ρ_{∗}(Y)ρ_{∗}(Z)E])
= B (ξ, ρ_{∗}(Y)ρ_{∗}(Z)E) = B (−ρ_{∗}(Y)ξ, ρ_{∗}(Z)E)
= B (−ρ_{∗}(Y)ξ, z) .

Then

A_{ξ}y = −ρ_{∗}(Y)ξ

and we obtain

ρ_{∗}(X)α_{E} (y, z) = −A_{α_{E}(y,z)}x.

This clearly shows that

(▽_{x}^{c}α) (y, z) = 0 ∀ x, y, z ∈ T_{E} (M_{E})

and completes the proof of the proposition. □ Lemma 14. The dimensions of the manifolds M and M_{E} are either both even or both odd and dim M_{E} < dim M. Proof. Let us assume that the dimension of M is even and let S be the unit sphere in T_{E}(M). Then dim S is odd and this implies that its Euler-Poincare characteristic vanishes χ (S) = 0. Let S_{E} be the unit sphere in T_{E} (M_{E}). The sphere S supports the action of the torus T ⊂ [K_{E}]_{e} ⊂ K_{E} (by the isotropy representation) and since S_{E} = F (T, S) we have χ (S_{E}) = 0. This obviously indicates that dim S_{E} is odd and in turn implies that dim M_{E} is even as we wanted to prove. The proof in the other case is similar. The inequality follows from orollary 7. □ Lemma 15. X [M_{E}] = F (T, X [M]). Proof. We work at the point E. We know that M_{E} is a totally geodesic submanifold of M with respect to the Riemannian connection and ▽^{c}-autoparallel. This implies that we have the following identities where the upper index M or M_{E} indicates that the object corresponds to the manifold M or to M_{E}.

If x, y, z ∈ T_{E} (M_{E}) and y^{∗} is any tangent vector field in M_{E} such that y^{∗} (E) = y we have

(i) ∇_{x}^{cM_{E}}y^{∗} = ∇_{x}^{cM}y^{∗}, (ii) ∇_{x}^{M_{E}}y^{∗} = ∇_{x}^{M}y^{∗}, (iii) D^{M_{E}} (y, z) = D^{M} (y, z) , (iv) α^{M_{E}} (y, z) = α^{M} (y, z).

These last two follow from (i) and (ii), and then we get

(v) α^{M_{E}} (x, D^{M_{E}} (y, z)) = α^{M} (x, D^{M} (y, z)).

Let now x ∈ X̂ [M_{E}] (for definition see (1) or [5, p.227]) then Ad (g) x = x ∀g ∈ T and

α^{M_{E}} (x, D^{M_{E}} (x, x)) = 0.

By (v) we have now

α^{M} (x, D^{M} (x, x)) = 0.

which shows that x ∈ F (T, X̂ [M]).

If, on the other hand we have x ∈ F (T, X̂ [M]) then Ad (g) x = x ∀g ∈ T which implies x ∈ V and since α^{M} (x, D^{M} (x, x)) = 0 we have by (v) α^{M_{E}} (x, D^{M_{E}} (x, x)) = 0 and then x ∈ X̂ [M_{E}]. This yields the equality of the lemma. □ Corollary 16. χ (X [M_{E}]) = χ (X [M]). Proof. This follows from the previous lemma and [5, p.234, (3.11)]. □

4. First part of the proof of the main theorem

We can give now a proof of (i) of Theorem 2.

Let M be our real flag manifold and j : M → 𝔭 its natural embedding. By Corollary 7 we may construct a proper submanifold M_{1} ⊂ M (M_{1} = M_{E} = [F (T, M)]_{E}) which, by Lemma 11, is ∇^{c}-autoparallel in M and its “induced” connection coincides with the canonical connection associated to the reductive decomposition 𝔠 = 𝔠_{E} ⊕ 𝔲 .

Furthermore there is a subspace 𝔮 ⊂ 𝔭 such that j (M_{1}) ⊂ 𝔮 and this is an embedding with parallel second fundamental form with respect to the canonical connection that we have in M_{1} (𝔮 = Q from Proposition 12). By the main result in [10] this is equivalent to the fact that M_{1} is a real flag manifold and j |M_{1} is its canonical embedding (this is full by the way in which we constructed the subspace 𝔮). Clearly this process has lowered the dimension of M into that of M_{1}. By Lemma 14 the dimension of M_{1} is even or odd in accordance with the dimension of M.

We may start now all over again with M_{1} instead of M and construct a new real flag manifold M_{2} and continuing in this manner we may obtain a sequence of proper submanifolds

M ⫌ M_{1} ⫌ M_{2} ⫌ … ⫌ M_{k},

all of which are real flag manifolds and having even or odd dimension according to that of M itself.

This process must stop and we need to find out where. To that end let us analyze what are the possibilities when we want to construct M_{s+1} from M_{s}. Let us first remark that since j |M_{s} is in fact the full canonical embedding of the real flag manifold M_{s}, Corollary 7 implies that if T_{s} is a maximal torus in the connected component of the identity of the isotropy group of E in M_{s} then

[F (T_{s}, M_{s})]_{E} ⫋ M_{s}.

Clearly we have now two possibilities, namely

\begin{matrix}1) & [F (T_{s}, M_{s})]_{E} & = & \{E\} \\ 2) & [F (T_{s}, M_{s})]_{E} & ⫌ & \{E\} \end{matrix}

and in any case, we define M_{s+1} = [F (T_{s}, M_{s})]_{E}.

In the first case (which happens only when the dimension of M is even) we have M_{s+1} = {E} , X [M_{s+1}] = ϕ and χ (X [M_{s+1}]) = 0. From Corollary 16 it follows that

χ (X [M]) = 0.

Let us consider now the situation (2).

If the dimension of M is even our process of construction M_{s+1}, …, M_{s+t} may continue until we reach a t such that dim M_{s+t} = 0 and then we are back to case (1) above.

If on the other hand dim M is odd, since dim M_{s} − dim M_{s+1} = 2r > 0, we could not end with dim M_{s+t} = 0 so our last possibility is dim M_{s+t} = 1 for some t. We have then a one-dimensional real flag manifold M_{s+t} and j |M_{s+t} is its full canonical embedding. It is known that for this embedding the second fundamental form is onto ([10]) and therefore the normal space of the embedding must be one-dimensional too. This means that the subspace 𝔮 ⊂ 𝔭 such that j (M_{s+t}) ⊂ 𝔮 , has dimension two and so M_{s+t} is just a curve contained in a plane. Then there is only one normal section which is obviously planar and in turn X [M_{s+t}] consists of a single point. Therefore χ (X [M_{s+t}]) = 1 and this yields χ (X [M]) = 1. This completes the proof of (i) of Theorem 2.

5. Lemmas for the complex case

Let V^{⊥} be the orthogonal complement of V = T_{E} (M_{E}) in T_{E} (M). It is clear that V^{⊥} is T -invariant.

Lemma 17. F (T, CP (T_{E} (M)^{c})) = CP (T_{E} (M_{E})^{c}) ∪ F (T, CP (V^{⊥c})).

Proof. For v ∈ T_{E} (M)^{c} , [v] means the equivalence class of v in the projective space CP (T_{E} (M)^{c}). We have [v] ∈ F (T, CP (T_{E} (M)^{c})) if and only if Cv is a subspace in T_{E} (M)^{c} invariant by the action of T.

Since

T_{E} (M)^{c} = T_{E} (M_{E})^{c} ⊕ V^{⊥c}

and T_{E} (M_{E})^{c} = F (T, T_{E} (M)^{c}) , any T -invariant complex 1-dimensional subspace is contained either in T_{E} (M_{E})^{c} or in V^{⊥c}. □

In Section 2 we gave the definition of a real flag manifold as orbit of E ∈ 𝔭 by the adjoint action of K, the analytic subgroup of G corresponding to 𝔨. Let 𝔤_{c} be the complexification of 𝔤 and σ the conjugation of 𝔤_{c} with respect to 𝔤. Since 𝔤 = 𝔨 ⊕ 𝔭 is a Cartan decomposition, there exists a compact real form 𝔤_{u} of 𝔤_{c} such that

σ𝔤_{u} ⊂ 𝔤_{u},
𝔨 = 𝔤 ∩ 𝔤_{u},
𝔭 = 𝔤 ∩ i𝔤_{u},
𝔤_{u} = 𝔨 ⊕ i𝔭.

Let G_{c} be the simply connected semisimple Lie group associated to 𝔤_{c}. Let G_{1} and G_{u} be the analytic subgroups of G_{c} corresponding to the subalgebras 𝔤 and 𝔤_{u} respectively. By [7, p. 152, 4 (ii)] they are both closed in G_{c} and by [7, p.256, 2.2 (iii)] G_{u} is also simply connected. Let K_{1} be the analytic subgroup of G_{c} corresponding to 𝔨. The group K_{1} is compact and clearly K_{1} ⊂ G_{1} ∩ G_{u}. Now Ad_{G_{1}} : G_{1} → G is a surjective analytic homomorphism and K = Ad_{G_{1}} (K_{1}).

On the other hand, if we take on i𝔭 the Euclidean metric induced by −B_{u}, the orbit M̂ = Ad_{G_{1}} (K_{1}) iE ⊂ i𝔭 is isometric to our space M. In fact, by [7, p. 180] if X, Y ∈ 𝔭 then

−B_{u} (iX, iY) = −B_{c} (iX, iY) = B_{c} (X, Y) = B (X, Y) .

Let M_{c} be the orbit of iE by Ad_{G_{u}} (G_{u}) ; it is obviously a complex flag manifold. Now consider on M_{c} the Riemannian metric induced by the inner product on 𝔤_{u} defined by −B_{u}. It is clear that M̂ is isometrically embedded in M_{c}.

Proposition 18. M̂ is totally geodesic in M_{c}.

Proof. Recall that σ denotes the conjugation of 𝔤_{c} with respect to 𝔤. Since σ is an automorphism of 𝔤_{u} and G_{u} is simply connected, σ induces an automorphism Θ of G_{u} defined by

Θ (exp (sX)) = exp (sσX) .

Then

Ad (Θ (exp (sX))) = exp (ad (sσX)) = σ ∘ exp (ad (sX)) ∘ σ

that is

Ad (Θ (g)) = σ ∘ Ad (g) ∘ σ, ∀g ∈ G_{u}.

Since iE ∈ i𝔭, σ (iE) = −iE and so

Ad (Θ (g)) iE = σ [Ad (g) (σ (iE))] = −σ (Ad (g) iE)

and since M_{c} = {Ad (g) iE : g ∈ G_{u}} , the automorphism Θ induces a well defined map ε : M_{c} → M_{c} by

ε (Ad (g) iE) = Ad (Θ (g)) iE.

Clearly

ε (p) = −σ (p) ∀p ∈ M_{c}

and hence ε is an isometry of M_{c} with the induced metric from −B_{u} on 𝔤_{u}.

Since M̂ ⊂ i𝔭 it is clear that M̂ is contained in [F (ε, M_{c})]_{iE} , the connected component of F (ε, M_{c}) containing iE. It also clear that ε_{∗iE} = ε and

T_{iE} (F (ε, M_{c})) = F (ε_{∗iE}, T_{iE} (M_{c})) .

Now

T_{iE} (M_{c}) = [𝔤_{u}, iE] = [𝔨, iE] ⊕ [i𝔭, iE] = T_{iE}(M̂) ⊕ 𝔴

where T_{iE}(M̂) ⊂ i𝔭 and 𝔴 ⊂ 𝔨.

Then we have

F (ε_{∗iE}, T_{iE} (M_{c})) = T_{iE}(M̂)

because any vector in T_{iE} (M_{c}) with non-zero component in 𝔴 is not fixed by ε_{∗iE}.

This proves that M̂ is open in [F (ε, M_{c})]_{iE} but since it is compact, it is also closed and so we have M̂ = [F (ε, M_{c})]_{iE}.

It is a well known fact that the fixed point set of an isometry in a Riemannian manifold is totally geodesic. This proves the assertion. □

Following previous notation, we use upper indexes M̂ or M_{c} to indicate that the object under consideration corresponds to M̂ or to M_{c}.

Corollary 19. i) αM̂ = α^{M_{c}} |M̂ , ii) DM̂ = D^{M_{c}} |M̂.

C. U. Sánchez et al.: Planar Normal Sections on the Natural Embedding … 525

Proof. (i) follows from the fact that M̂ is totally geodesic in M_{c}. (ii) is a consequence of Proposition 5. □ The above corollary yields that

αM̂(x, DM̂ (x, x)) = α^{M_{c}} (x, D^{M_{c}} (x, x)) ∀x ∈ T_{iE}(M̂). (2)

The complexifications of α and D are

α^{c} (x_{1} + iy_{1}, x_{2} + iy_{2}) = α (x_{1}, x_{2}) − α (y_{1}, y_{2}) + i [α (y_{1}, x_{2}) + α (x_{1}, y_{2})]
D^{c} (x_{1} + iy_{1}, x_{2} + iy_{2}) = D (x_{1}, x_{2}) − D (y_{1}, y_{2}) + i [D (y_{1}, x_{2}) + D (x_{1}, y_{2})] .

By definition X_{c}[M̂] (resp. X_{c} [M_{c}]) is the algebraic subvariety of CP (T_{iE}(M̂)^{c}) (resp. CP (T_{iE} (M_{c})^{c})) defined by the equation

(αM̂)^{c}(z, (DM̂)^{c} (z, z)) = 0

(resp. (α^{M_{c}})^{c} (z, (D^{M_{c}})^{c} (z, z)) = 0)

for z = x + iy ∈ T_{iE}(M̂)^{c} (resp. z = x + iy ∈ T_{iE} (M_{c})^{c}).

Now a straight forward computation shows that Lemma 20. (αM̂)^{c}(z, (DM̂)^{c} (z, z)) = (α^{M_{c}})^{c}(z, (D^{M_{c}})^{c} (z, z)) for every z = x + iy ∈ T_{iE}(M̂)^{c} ⊂ T_{iE} (M_{c})^{c}. Then we have Lemma 21. X_{c}[M̂] = CP (T_{iE}(M̂)^{c}) ∩ X_{c} [M_{c}]. Let us set M̂_{iE} = [F (T, M̂)]_{iE} and V̂ = T_{iE}(M̂_{iE}). The reader can immediately verify that the proof of Lemma 17 translates mutatis mutandis into a proof of the following Lemma 22. F (T, CP (T_{iE}(M̂)^{c})) = CP (T_{iE}(M̂_{iE})^{c}) ∪ F (T, CP (V̂^{⊥c})). We have now Proposition 23. F (T, X_{c}[M̂]) = X_{c}[M̂_{iE}] ∪ [F (T, CP (V̂^{⊥c})) ∩ X_{c} [M_{c}]]. Proof. By Lemma 21

X_{c}[M̂] = CP (T_{iE}(M̂)^{c}) ∩ X_{c} [M_{c}]

and since V̂^{⊥c} ⊂ T_{iE}(M̂)^{c} we have

CP (V̂^{⊥c}) ⊂ CP (T_{iE}(M̂)^{c}).

Then

CP (V̂^{⊥c}) ∩ X_{c} [M_{c}] ⊂ X_{c}[M̂]

and furthermore

F (T, CP (V̂^{⊥c})) ∩ X_{c} [M_{c}] ⊂ F (T, X_{c}[M̂]).  (3)

Besides, it is clear that equality (v) from the proof of Lemma 15 holds for the manifolds M̂_{iE} and M̂ therefore

X_{c}[M̂_{iE}] ⊂ F (T, X_{c}[M̂])  (4)

and then (3) and (4) prove one of the required inclusions.

Instead of proving the other one, we rather see that

F (T, X_{c}[M̂]) ⊂ X_{c}[M̂_{iE}] ∪ F (T, CP (V̂^{⊥c})).

Since by definition

X_{c}[M̂] ⊂ CP (T_{iE}(M̂)^{c})

it is clear that

F (T, X_{c}[M̂]) ⊂ F (T, CP (T_{iE}(M̂)^{c})) =
= CP (T_{iE}(M̂_{iE})^{c}) ∪ F (T, CP (V̂^{⊥c}))

by Lemma 22.

Then it suffices to show that

F (T, X_{c}[M̂]) ∩ CP (T_{iE}(M̂_{iE})^{c}) ⊂ X_{c}[M̂_{iE}]

and in fact it is enough to see that

X_{c}[M̂] ∩ CP (T_{iE}(M̂_{iE})^{c}) ⊂ X_{c}[M̂_{iE}]

and this is clear.

Let us consider now the sets A, B and C defined as follows

A = X_{c}[M̂_{iE}] ∪ [F (T, CP (V̂^{⊥c})) ∩ X_{c} [M_{c}]]
B = F (T, X_{c}[M̂])
C = X_{c}[M̂_{iE}] ∪ F (T, CP (V̂^{⊥c})).

Then we have the inclusions

A ⊂ B ⊂ C

and intersecting with X_{c} [M_{c}] we observe that

A ∩ X_{c} [M_{c}] = A
B ∩ X_{c} [M_{c}] = B
C ∩ X_{c} [M_{c}] = A.

Then

A = B

and the proof of the proposition is complete. □

Since X_{c}[M̂_{iE}] and F (T, CP (V̂^{⊥c})) are clearly disjoint, it follows that

χ(F (T, X_{c}[M̂])) = χ(X_{c}[M̂_{iE}]) + χ(F (T, CP (V̂^{⊥c})) ∩ X_{c} [M_{c}]).

Now we have the following crucial fact

Proposition 24. χ(F (T, CP (V̂^{⊥c})) ∩ X_{c} [M_{c}]) = χ(F (T, CP (V̂^{⊥c}))).

Proof. For the manifold M_{c} = G_{u}/ (G_{u})_{iE} we may consider the center S of the isotropy group (G_{u})_{iE}. The group S plays an essential role in the proof of the main result in [5]. It acts naturally on T_{iE} (M_{c}) and then on CP (T_{iE} (M_{c})^{c}) (see [5, p.231–234]) . There, T_{iE} (M_{c})^{c} is denoted by 𝔐^{c} and we shall adopt here the same simpler notation.

Since V̂^{⊥c} ⊂ 𝔐^{c}, we have

F (T, CP (V̂^{⊥c})) ⊂ CP (𝔐^{c}) .

Our torus T is maximal in K_{E} ⊂ K but, as we have observed, K = Ad_{G_{1}} (K_{1}) and K_{1} ⊂ G_{1} ∩ G_{u}. There is a torus T_{1} ⊂ K_{1} such that Ad_{G_{1}} (T_{1}) = T (in fact we may take T_{1} = [Ad_{G_{1}}^{−1} (T)]_{e} which is clearly compact, connected and abelian).

Now T_{1} acts on M̂ exactly as T and then

F (T_{1}, CP (V̂^{⊥c})) = F (T, CP (V̂^{⊥c})).

Furthermore F (T, CP (V̂^{⊥c})) is invariant by the action of S on CP (𝔐^{c}) because S commutes with T_{1}. Then it makes sense to consider the submanifold F (S, F (T, CP (V̂^{⊥c}))).

We have now the following obvious identity

F (S, F (T, CP (V̂^{⊥c}))) = F (S, F (T, CP (V̂^{⊥c}))) ∩ F (S, CP (𝔐^{c})) .  (5)

Now, part (ii) of Proposition (3.10) in [5, p.234] says, in our notation,

F (S, CP (𝔐^{c})) = F (S, X_{c} [M_{c}])

and hence we may replace it in (5) obtaining

F (S, F (T, CP (V̂^{⊥c}))) = F (S, F (T, CP (V̂^{⊥c}))) ∩ F (S, X_{c} [M_{c}]) =
= F (S, [F (T, CP (V̂^{⊥c})) ∩ X_{c} [M_{c}]]).

This proves

χ(F (S, F (T, CP (V̂^{⊥c})))) = χ(F (S, [F (T, CP (V̂^{⊥c})) ∩ X_{c} [M_{c}]])).

Now we apply Theorem (3.11) in [5, p.234] and obtain our proposition. □

Now, from Propositions 23 and 24 we obtain (again by [5, p.234, (3.11)])

Corollary 25. χ(X_{c}[M̂]) = χ(X_{c}[M̂_{iE}]) + χ(F (T, CP (V̂^{⊥c}))).

Let us consider now the diagram

M_{E} ↪ M ↪ 𝔭
↓ f ↓ f ↓ f
M̂_{iE} ↪ M̂ ↪ i𝔭

where f(X) = iX is clearly an isometric isomorphism from 𝔭 to i𝔭 and therefore it induces isometries on the other levels because we are always taking the induced metric on the submanifolds. Furthermore, since f_{∗} takes planar normal sections into planar normal sections, from our previous corollary, we get

Corollary 26. χ (X_{c} [M]) = χ (X_{c} [M_{E}]) + χ (F (T, CP (V^{⊥c}))).

6. Second part of the proof of the main theorem

We are, finally, in position to prove part (ii) of Theorem 2. By Corollary 26 we have

χ (X_{c} [M]) = χ (X_{c} [M_{E}]) + χ (F (T, CP (V^{⊥c}))) .

If, by inductive hypothesis, we know that

χ (X_{c} [M_{E}]) = χ (CP (T_{E} (M_{E})^{c}))  (6)

then

χ (X_{c} [M]) = χ (CP (T_{E} (M_{E})^{c})) + χ (F (T, CP (V^{⊥c}))) .

By Lemma 17 we have now

χ (X_{c} [M]) = χ (F (T, CP (T_{E} (M)^{c})))

and then again by [5, p. 234, (3.11)]

χ (X_{c} [M]) = χ (CP (T_{E} (M)^{c})) .

The proof is then reduced to get identity (6) and to that end we have to proceed as in Section 4 setting M_{1} = M_{E} = [F (T, M)]_{E} and constructing our sequence of proper submanifolds

M ⫌ M_{1} ⫌ M_{2} ⫌ … ⫌ M_{k},

all of which are real flag manifolds and having even or odd dimension according to that of M itself.

This process must stop and, as we noticed in Section 4, if our last manifold is M_{k} then we have the following possibilities

1) M_{k} = {E} (dim M even)
2) dim M_{k} = 1 (dim M odd) .

If we have M_{k} = {E} then X [M_{k}] = ϕ and also X_{c} [M_{k}] = ϕ which clearly implies χ (X_{c} [M_{k}]) = 0. Furthermore in this case T_{E} (M_{k})^{c} = {0} and then CP (T_{E} (M_{k})^{c}) = ϕ.

This yields χ (CP (T_{E} (M_{k})^{c})) = 0 and so the identity (6) holds trivially in the last step of our “reversed ” induction process.

On the other hand, if we have the second possibility for M_{k} then, as was indicated in Section 4, we have that X [M_{k}] consists of a single point. Since dim M_{k} = 1 we have T_{E} (M_{k}) ≃ R and of course (T_{E} (M_{k}))^{c} ≃ C. Then CP (T_{E} (M_{k})^{c}) consists of a single point and clearly so does X_{c} [M_{k}].

Then we have

χ (X_{c} [M_{k}]) = 1 = χ (CP (T_{E} (M_{k})^{c}))

and the identity (6) also holds trivially in the last step of our “reversed” induction process if the dimension of M is odd.

This completes the proof of Theorem 2. □

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Received January 20, 2000