CHAPTER IV

THE REAL GRASSMANNIANS

1. The real Grassmannians

The involution of X, corresponding to the change of orientation of an m-plane of F, is an isometry of X. The group of isometries of X generated by , which is of order 2, acts freely on X and we may consider the Riemannian manifold endowed with the Riemannian metric gY induced by g. The natural projection is a two-fold covering. We identify Y with the real Grassmannian of all m-planes in F. When m = 1, the Grassmannian is the projective space of F.

Let VY and WY be the vector bundles over Y whose fibers at the point y Y are equal to Vx and Wx, respectively, where x is one of the points of X satisfying (x) = y. Then the tangent space TY,y of Y at y Y is identified with

For x X, the tangent space T(x) is equal to it is easily verified that the mappin is equal to the identity mapping of A vector field on X is even (resp. odd) with respect to the involution if = (resp. = -). We say that a symmetric p-form u on X is even (resp. odd) with respect to if where = 1 (resp. = -1). Such a form u is even if and only if we can write where is a symmetric p-form on Y . If E is a sub-bundle of invariant under the isometry , there exists a unique sub-bundle EY of such that, for all x X, the isomorphism where y = induces by restriction an isomorphism : E symmetric p-form u on Y is a section of EY if and only if the even symmetric p-form on X is a section of E.

Throughout the remainder of this section, we suppose that n 1. The curvature R of the Riemannian manifold (X, g) is determined by

We denote by the sub-bundle of which is the orthogonal complement, with respect to the scalar product induced by g1, of the line bundle {g1} generated by the section g1 of . Similarly, we denote by Wthe sub-bundle of which is the orthogonal complement of the line bundle {g2} generated by the section g2. We consider the sub-bundles

We also consider the Grassmannians = and the natural projection : and the involution of Let be the canonical vector bundle of rank n over whose fiber at is the subspace of F determined by the oriented n-plane a, and let be the vector bundle of rank m over whose fiber over a Xis the orthogonal complement of in F. As above, we identify the tangent bundle of Xwith the bundle the scalar product on F induces Riemannian metrics on and . There is a natural diffeomorphism

sending an m-plane of F into its orthogonal complement. When m = n, this mapping is an involution of in this case, we say that a symmetric p-form u on Y is even (resp. odd) if where = 1 (resp. = -1).

Now suppose that the vector space F is oriented and let be a unit vector of which is positive with respect to the orientation of F. The oriented m-plane x X gives us an orientation of Vx, which in turn induces an orientation of Wx: if {v1, . . . , vm} is a positively oriented orthonormal basis of Vx, then the orientation of Wx is determined by an orthonormal basis {w1, . . . ,wn} of Wx satisfying

endowed with the Riemannian metrics g and gY induced by the standard Euclidean scalar product of Throughout the remainder of this section, we also suppose that m + n 3.

The group SO(m + n) acting on sends every oriented m-plane into another oriented m-plane. This gives rise to an action of the group SO(m + n) on X. In fact, the group SO(m + n) acts transitively on the Riemannian manifold (X, g) by isometries. The isotropy group of the point x0 of X corresponding to the vector e is the subgroup of SO(m + n) consisting of the matrices

If B is the Killing form of the restriction to of the scalar product -B is invariant under the adjoint action of K and therefore induces an SO(m + n)-invariant metric on the homogeneous space SO(m + n)/K. Endowed with this metric g0, the manifold SO(m + n)/K is a symmetric space of compact type of rank min(m, n). It is easily verified that Thus is an isometry from the symmetric space SO(m + n)/K, endowed with the metric

to X; henceforth, we shall identify these Riemannian manifolds by means of this SO(m + n)-equivariant isometry. From Lemma 1.21, we again obtain the equality (4.3). Moreover, the symmetric space SO(m + n)/K is irreducible unless m = n = 2. On the other hand, we shall see below that the Grassmannian is not irreducible and is in fact isometric to a product of 2-spheres (see Proposition 4.3).

We now suppose that m = n = 3. If we identify the vector bundle with V and the vector bundle with W by means of the scalar products g1 on V and g2 on W, the Hodge operators

when m = n = 3. We recall that the Grassmannian is isometric to the symmetric space SU(4)/SO(4) (see [36, p. 519]). Thus by Lemma 2.41, we know that the space Hom vanishes for the space when m, n 3 and m + n 7, and that Hom is one-dimensional for the space as we have just seen, both of these assertions are also consequences of Lemma 4.1.

Let be the element of the dual of the group SO(m + n) which is the equivalence class of the irreducible SO(m + n)-module . We denote by K the SO(m + n)-module of all Killing vector fields on X and by KC its complexification. The irreducible symmetric space X is not equal to a simple Lie group. Thus according to (2.27), we know that the SO(m+n)module is irreducible and is equal to KC. When m + n 7, by (4.16) the Frobenius reciprocity theorem tells us that

(ii) If m = n = 3, then E(X) is an irreducible SO(6)-module isomorphic to the Lie algebra and is equal to the SO(6)-submodule

When m + n 7, the vanishing of the space E(X) is also given by Theorem 1.22 (see Koiso [41] and [42]).

For the remainder of this section, we suppose that m = n 2. Then the isometry of the Grassmannian is an involution. The group of isometries of Y generated by , which is of order 2, acts freely on Y and we may consider the Riemannian manifold equal to the quotient Y/endowed with the Riemannian metric induced by g. The natural projections and are two-fold and four-fold coverings, respectively. The action of the group SO(2n) on Y passes to the quotient In fact, SO(2n) acts transitively on and it is easily verified that the isotropy group of the point is equal to the subgroup of SO(2n) generated by and the matrix

of SO(2n). In fact, is a symmetric space of compact type of rank n. When n 3, it is irreducible and equal to the adjoint space of X and of Y . On the other hand, when n = 2, it is not irreducible, and we have the following result, whose proof appears below in 9, Chapter V:

PROPOSITION 4.3. The symmetric space ( endowed with the Riemannian metric which is the product of the metrics of constant curvature 1 on each factor is isometric to the Grassmannian endowed with the Riemannian metric Y ).

The following proposition is a direct consequence of Propositions 4.3 and 10.2, and of Theorem 2.23,(ii).

PROPOSITION 4.4. The maximal flat Radon transform for functions on the symmetric space G

R2 ,2 is injective.

The notion of even or odd tensor on Y (with respect to the involutive isometry defined here coincides with the one considered in 4, Chapter II. In fact, a section u of over Y is even if and only if we can write where is a symmetric p-form on Y . Lemma 2.17 gives us the following result:

LEMMA 4.5. A symmetric p-form u on satisfies the Guillemin condition if and only if the even symmetric p-form on satisfies the Guillemin condition.

2. The Guillemin condition on the real Grassmannians

Let m, n 1 be given integers. In this section, we again consider the real Grassmannians X = and Y = endowed with the metrics g and gY , and the natural Riemannian submersion and continue to identify the tangent bundle T of X with the vector bundle as in 1.

of closed totally geodesic submanifolds of Indeed, using the above description of the mappin at (x1, . . . , xr), we see that the tangent spaces of these two submanifolds of at x are equal. From the formula for the curvature of we infer that ExpxV 1 W1 is a totally geodesic submanifold of and a globally symmetric space. Clearly, the sub-manifold possesses these same properties. In fact, the subgroup SO(m+n, V1) of SO(m+n) consisting of all elements of SO(m+n) which preserve the subspace V1 and which are the identity on the orthogonal complement of V1 acts transitively on these submanifolds by isometries. These various observations yield the relation (4.21), which in turn gives us the equality

From the above observations and the equalities (4.21) and (4.22), we obtain:

(ii) if u is an even (resp. odd) symmetric form on X, the form iu on Z is even (resp. odd);

(iii) if h is a section of the sub-bundle E of over X, then ih is a section of the sub-bundle EZ of .

If the subspaces Vj are all 2-dimensional and the integers pj are all equal to 1, then the images of the mappings and are totally geodesic flat r-tori of and In particular, when m n and r = m, these images are maximal flat totally geodesic tori of and and all maximal flat totally geodesic tori of and arise in this way. On the other hand when n < m and r = n + 1, if the subspaces Vj are 2-dimensional and pj = 1 for and if = m - n, then the images

m-From Lemma 2.17, or from the commutativity of diagram (4.23) and the above remarks concerning totally geodesic flat tori of the Grassmanni-ans, we obtain the following result:

LEMMA 4.7. A symmetric p-form u on Y satisfies the Guillemin condition if and only if the even symmetric p-form on X satisfies the Guillemin condition.

of Y , which is isometric to If u is a symmetric p-form on Y satisfying the Guillemin condition, then the restriction of u to satisfies the Guillemin condition.

holds. Now we suppose that u satisfies the Guillemin condition; then the above integral vanishes, and so the function f on the real projective space of dimension 2 satisfies the zero-energy condition. The injec-tivity of the X-ray transform for functions on the real projective space of dimension 2, given by Theorem 2.23,(ii), tells us that the function f vanishes. From the equality f(x0) = 0, we infer that the restriction of u to satisfies the Guillemin condition.

The following proposition is a generalization of Lemma 5.3 of [23].

From Proposition 4.10 and Lemma 4.6, by means of the isometries : GR ,p we deduce the following:

PROPOSITION 4.12. Let y be a point of the real Grassmannian Y = Let be a closed totally geodesic submanifold of Y isometric to the real Grassmannian which can be written in the form Expy where is a q-dimensional subspace of V and is an r-dimensional subspace of Assume either that 2 m < n and r = n, or that 2 n < m and q = m. If u is a symmetric p-form on Y satisfying the Guillemin condition, then the restriction of u to satisfies the Guillemin condition.

When the following proposition is a consequence of Propositions 4.11 and 4.12 and Lemma 4.6. If in the following proposition, the submanifold of Y considered there has the same rank as Y , and in this case the conclusion of the proposition is immediate.

PROPOSITION 4.13. Let y be a point of the real Grassmannian Y = with 2 m < n. Let Y be a closed totally geodesic submanifold of Y isometric to the real Grassmannian which can be written in the form where is a q-dimensional subspace of If u is a symmetric p-form on Y satisfying the Guillemin condition, then the restriction of u to satisfies the Guillemin condition.

From Proposition 4.12 and the injectivity of the X-ray transform for functions on a real projective space, we now obtain the following proposition, which is also given by Theorem 2.24.

PROPOSITION 4.14. For m, n 2, with the maximal flat Radon transform for functions on the real Grassmannian is injective.

PROOF: Let Z be the closed totally geodesic submanifold of Y equal ,n; clearly to Expy which is isometric to the real Grassmannian is a totally geodesic submanifold of Z. Let be a 1-form on Y satisfying the Guillemin condition. By Proposition 4.12, the restriction of to the submanifold Z satisfies the zero-energy condition; therefore so does the restriction of to the submanifold By Theorem 3.26, we know that is exact.