Leveringstid: Sendes innen 7 virkedager. Om boka. Avbryt Send e-post. Morse Theory. AM , Volume 51 John Milnor One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Nicholas M. Michael Harris og Professor Richard Taylor. Amnon Neeman. Karl Rubin. Eric M.

Friedlander , Andrei Suslin og Vladimir Voevodsky. Demetrios Christodoulou. Jacek Graczyk og Grzegorz Swiatek. Curtis T. Michael Rapoport og Thomas Zink. Christine Lescop. Paulo Cordaro og Francois Treves. Louis H. Kauffman og Sostenes Lins. Pierre Deligne og G. Daniel Mostow. William Fulton. James Eells og Andrea Ratto. Theorem 2. The adjoint space of the symmetric space X is the symmetric space which admits X as a Riemannian covering and is itself not a Riemannian covering of another symmetric space. In [34], Grinberg generalized Theorem 2.

By Proposition 2. Hence from Proposition 2. Let X be an irreducible symmetric space of compact type. If X is rigid in the sense of Guillemin, then X is equal to its adjoint space. Then according to Lemma 2. Thus if F is a sub-bundle of TC resp. Since the mapping 2. Thus we have 2. From the above equalities, we obtain the inclusions 2.

We obtain the following two propositions from the inclusions 2. Let X, g be a symmetric space of compact type. By Lemmas 2. Clearly, the space X is rigid in the sense of Guillemin resp. Radon transforms and harmonic analysis 55 holds resp. Thus according to Proposition 2. On the other hand, according to Proposition 2. Lie algebras Let g be a complex semi-simple Lie algebra.

Irreducible symmetric spaces 59 Lemma 2. Let g be a complex semi-simple Lie algebra. Let G be a compact connected, semi-simple Lie group, whose Lie algebra we denote by g0. A complex G-module V can be viewed as a g0 -module and so the Casimir element of g0 operates on V ; if V is an irreducible G-module, the Casimir element of g0 acts by the scalar c V on V. Let G be a compact connected, semi-simple Lie group. Let V1 and V2 be irreducible complex G-modules. We denote by S k p the k-th symmetric product of p and by S02 p the K-submodule of S 2 p consisting of those elements of S 2 p of trace zero with respect to the Killing form of g.

If X is an irreducible Hermitian 60 II. If X is an irreducible symmetric space which is not Hermitian, then we have 2. We suppose throughout the remainder of this section that X is an irreducible symmetric space. In fact, by formula 1. Moreover, by Proposition 2. Hence by Proposition 2. Proof: Let u be an element of NC resp. Since NC resp. Proof: Here we shall use Lemma 2. The desired result is a direct consequence of formulas 1.

Irreducible symmetric spaces 63 We now further assume that X is of type I, i. This observation and the above remarks concerning the Lichnerowicz Laplacians, together with the Frobenius reciprocity theorem, give us the following result: Lemma 2. Let X, g be an irreducible symmetric space of compact type which is not equal to a simple Lie group. Since the G-module KC is isomorphic to g, we see that 2. If X is not a Hermitian symmetric space, according to the second equality of 2. If X is a Hermitian symmetric space, by 2. Let X, g be an irreducible symmetric space of compact type, which is not equal to a simple Lie group or to the sphere S 2.

Hence the equality 2. The other assertions of the proposition then follow from the second equalities of 2. In [42], Koiso also showed that the assertions of the previous proposition also hold when the irreducible space X is a simple Lie group. The following lemma is stated without proof by Koiso see Lemma 5.

Let X, g be a simply-connected irreducible symmetric space of compact type which is not equal to a simple Lie group. Therefore from Proposition 2. From the relations 2. Let d be the integer which is equal to 1 when X is a Hermitian symmetric space and equal to 0 otherwise. According to 2. Since its G-submodule KC is isomorphic to g, from Propositions 2. According to Proposition 2. Let F be a family of closed connected totally geodesic surfaces of X which is invariant under the group G. Then the set NF resp.

According to for resp. Let F be a G-invariant family of closed connected totally geodesic surfaces of X with positive constant curvature. Then the following assertions are equivalent: i The symmetric 2-form h belongs to L F. Proof: By Lemma 1. Now suppose that assertion ii holds. If gY is the Riemannian metric on Y induced by g, by Proposition 1. Since B a consequence of Proposition 1. Thus assertion ii is true.

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Finally, assertion iii is a direct consequence of i and ii. Suppose that the relation 1. According to the equality 2. By the relation 1. Thus we have proved assertion i. Then we know that the space N2 resp. Assertion ii resp. Criteria for the rigidity of an irreducible symmetric space 71 connected totally geodesic submanifolds of X.

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## Annals of Mathematics Studies

Suppose that 2. Proof: In view of Proposition 2. Proof: Under the hypotheses of i resp. Then assertions i and ii of Theorem 2. Thus according to Theorem 2. We again assume that X is an arbitrary symmetric space of compact type. From the relation 2. According to formulas 1. We may consider X as the quotient of the space Rn endowed with the Euclidean metric g0. Clearly, a tensor on Rn which is invariant under the group of all translations of Rn induces a tensor on X. Let x1 ,. Clearly, 76 III. According to formula 1. Let h be a symmetric 2-form on X.

Finally, suppose that iii holds. Then according to 3. Formula 3. From Proposition 3. Other proofs of these results are given in [2] and [15] see Proposition Lemma 3. From these last two observations, we obtain the desired result. As we saw above, we have the decomposition 3. From Lemma 3. The following proposition is due to Michel [46]. Proposition 3.

Thus the function f is constant and therefore vanishes. Flat tori 81 Proposition 3. Theorem 3. We now simultaneously prove Theorems 3. If gY is the metric on Y induced by g, Proposition 3.

According to Lemma 3. Theorems 3. The next theorem, which generalizes both of these theorems, was proved by Michel [46] when the integer p is equal to 0, 1 or an odd integer and by Estezet [12] in all the other cases. The next result is given by Proposition 1 of [20]. Let X, g be a Riemannian manifold.

The projective spaces 83 According to Lemma 1. Therefore by Proposition 3. The projective spaces In the remainder of this chapter, we shall consider the symmetric spaces of compact type of rank one. The following two theorems are consequences of Theorems 3. Let X be a symmetric space of rank one, which is not isometric to a sphere. Most of the results described in the remainder of this section are to be found in Chapter 3 of [5].

The projection 3. We see that the projection 3. We identify p0 with the vector space Kn and, in particular, the element 3. In fact, the metric g on complex manifold CPn is the Fubini-Study metric of constant holomorphic curvature 4 see Example On the other hand, 88 III.

The following result is a direct consequence of Proposition 3. Suppose that K is equal to C or H. Proof: By Proposition 3. Therefore, from Proposition 3. An analogue of Proposition 3. The real projective space 89 the Cayley algebra gives rise to closed totally geodesic submanifolds of X isometric to HP2. I in Chapter III of [4]. The X-ray transform for functions on a symmetric space of compact type of rank one, which is not isometric to a sphere, is injective. Proof: According to Proposition 3. The desired result is then a consequence of Theorem 2.

The real projective space 91 Proposition 3. Proof: Let h be a symmetric 2-form on X satisfying the zero-energy condition. We know that the relation 1. From Theorem 2. According to Proposition 3. Therefore by the relation 1. According to Theorem 1. The proof of Proposition 3.

We suppose that X, g is the real projective plane RP2 , g0. The following result given by Lemma 2. Let X be the real projective plane RP2. The lemma is now a consequence of 3. The real projective space 93 Lemma 3. Proof: According to Theorem 2. Then Proposition 1. Now Proposition 3. Our approach to the rigidity questions, which led us to the criteria of Theorem 2. The correspondence between the arguments given here in the case of the real projective plane and those used in the case of the complex quadric is pointed out in [30].

The following result is due to Michel [47]. In the case of the real projective plane RP2 , according to Proposition 2.

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In fact, the result given by Theorem 3. Then the desired result for X is a consequence of Proposition 2. The following result due to Bailey and Eastwood [1] generalizes both Theorems 3. We have seen that g is the Fubini-Study metric of constant holomorphic curvature 4. We denote by J the complex structure of X. According to 3. I and C.

The group of all diagonal matrices of G is a maximal torus of G and of K. According to the branching law for G and K see Proposition 3. According to the commutativity of diagram 3. Now let k be a given positive integer. From the relations 3. Let X be the complex projective space CP1. Then by Propositions 2. The rigidity of the complex projective space From formula 3.

We now introduce various families of closed connected totally geodesic submanifolds of X. The rigidity of the complex projective space while a surface belonging to the family F2 is isometric to the complex projective line with its metric of constant curvature 4. From these observations and the relations 3.

In [13, Lemma 8. Since the space X is simply connected, from Proposition 3. Proof: By Lemma 3. By Proposition 3. The equivalence of assertions i and iii of the following theorem is originally due to Michel [45]. The rigidity of the complex projective space Theorem 3. The following assertions are equivalent: i The symmetric 2-form h belongs to L F2. Proof: From the exactness of sequence 1. From Proposition 2. Consequently, we have: Lemma 3. From Theorem 3. According to the second formula of 1. Other proofs of Theorem 3.

We remark that the equivalence of assertions i and iii of Theorem 3. The rigidity of the complex projective space We now present an outline of the proof of Theorem 3.

Since X is an irreducible Hermitian symmetric space, according to 2. According to these observations, in order to prove the exactness of the sequence 3. Before proceeding to the description of another proof of Theorem 3. According to Theorem 3. The simplicity of the preceding proof, which is based on a remark of Demailly see [18] , rests on the correct interpretation of the bundle F. For symmetric 2-forms, the analogue of the bundle F of the preceding proof is the bundle N1 , and its interpretation is given by Proposition 3. We now describe the proof of Theorem 3.

The following two results are proved in [18]. The rigidity of the complex projective space Lemma 3. Let h be a symmetric 2-form on X satisfying the zero-energy condition. Thus by Proposition 3. Then Lemma 1. Hence the equality 3. By Lemma 3. Therefore without loss of generality, we may assume that the function f belongs to H1. Then Lemma 3. Let F1 be the family of all closed connected totally geodesic surfaces of X which are isometric either to the real projective plane with its metric of constant curvature 1 or to the sphere S 2 with its metric of constant curvature 4.

Let F2 be the family of all closed connected totally geodesic submanifolds of X isometric to the projective plane CP2. We verify that every surface belonging to the family F1 is contained in a submanifold belonging to the family F2 see Proposition 3. The following proposition can be proved by means of computations similar to the ones used in [14] to prove Proposition 3. The other projective spaces Proposition 3. We remark that the equalities of the preceding proposition imply that the relation 1. Thus we may apply Theorem 2. Proof: According to Propositions 3.

By Theorem 3. The latter theorem now gives us the desired result. The equivalence of assertions i and iii of Theorem 3. Then we have a natural isomorphism of vector bundles 4. The isomorphism 4. F is the projective space of F. Clearly, the vector bundle is a sub-bundle of E; then from the decomposition 4. The real Grassmannians sending an m-plane of F into its orthogonal complement. Clearly the diagram 4. All the vector bundles appearing in the decomposition 4.

Thus G m,n n,m spaces. From 4. Then by 4. From the commutativity of the diagram 4. Lemma 4. From Lemma 4. Thus by Lemma 2. The irreducible symmetric space X is not equal to a simple Lie group. Thus according to 2. G G 2,2 2,2 The following proposition is a direct consequence of Propositions 4. Proposition 4. Let V1 ,.

We have the equality 4. These various observations yield the relation 4. It follows that, if u is an even resp. The Guillemin condition on the real Grassmannians Lemma 4. If the restriction of h to an arbitrary submanifold of the family F vanishes, then h vanishes. It follows that h vanishes. Let u be a symmetric p-form on Y. The following proposition is a generalization of Lemma 5. We now consider the totally geodesic imbedding 4. Let u be a symmetric p-form on Y satisfying the Guillemin condition.

According to Lemma 4. Outline This chapter is devoted to the complex quadric which plays a central role in the rigidity problems. The involutions of the tangent spaces of Qn , which arise from the second fundamental form of the quadric, allow us to introduce various objects and vector bundles on Qn. In the next section, we describe the tangent spaces of various families of totally geodesic submanifolds of Qn and present results concerning the spaces of tensors of curvature type which vanish when restricted to some of these families.

Moreover, by 5. According V. The complex quadric viewed as a complex hypersurface We begin by recalling some results of Smyth [49] see also [21]. We denote by S the bundle of unit vectors of this normal bundle. Using formulas 3. From 5. According to this last remark and formulas 5. Then by 5. Then we have the equality 5. According to 5. On the other hand, formula 3. We equate the normal and tangential components of the right-hand sides of 5. From this last remark, the equality 5. Using 5. Using the equalities 5.

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We thus obtain the orthogonal decompositions 5. Clearly, we have 5. The bundles of the decomposition 5. Thus if F is one of the vector bundles appearing in the right-hand side of the decomposition 5. Then from 5. By means of the equation 5. It follows that 5. By means of 5. The image a of the point 1, 0,. From this last relation and the decomposition 5.

## Radon Transforms and the Rigidity of the Grassmannians (AM-156) (Annals of Mathematics Studies)

The equality 5. The relations 5. According to the commutativity of the diagram 5. On the other hand, the equalities 5. The complex quadric and the real Grassmannians From the above inclusions, the relations 5. We now suppose that n is even. By formulas 3. The decomposition 5.

By means of this frame and the relation 5. The following result is given by Lemma 4. Lemma 5. Let X be the quadric Q4. From formulas 1. By means of the decomposition 5.

B We now introduce various families of closed connected totally geodesic submanifolds of X. According to [10], if F is generated by the family A2 resp. Therefore such a submanifold Z is isometric to a sphere of constant curvature 2 see also [10] ; moreover, by Lemma 4. Moreover, according to [10] this surface is isometric to the real projective plane RP2 of constant curvature 1.

Moreover, this submanifold is isometric to the complex projective plane CP2 of constant holomorphic curvature 4. Clearly, V. Using the formula 1. The following three lemmas are proved in [21]. Then the following assertions hold: of the vector bundle N i We have 5. Then the equality 5. Proof: Let x be a point of X and u be an element of N1,x. Hence, by polarization we obtain the equality 5. The desired result is now a consequence of the second equality of 5. Let x be a point of X and u be an element of N2,x. Therefore according to the second formula of 5.

By the equality 5. This inclusion and 5. By Lemma 5. Hence according to 5. Thus we have proved assertion ii. The following two propositions are direct consequences of Proposition 5. In fact, Proposition 5. Multiplicities Proposition 5. Proposition 5. Using the relations 1. From these results, the relation 1. By methods similar to those used in [21] to prove Lemmas 5. The highest weight of an irreducible G-module resp.

The equivalence class of such a G-module resp. K linear forms on t. Since j belongs to the center of K, this decomposition of p is invariant under the action of K on p. We thus obtain the K-invariant decomposition 5. We consider Vx0 as an O 2 -module and Wx0 as an O n -module. Hence from the last of the equalities 5. Hence from the previous discussion, we obtain the following result: V. From Lemma 5. According to the decomposition 5. From these remarks, Lemma 5. Since the symmetric space Y is irreducible, is not equal to V. We have thus proved the following result: Proposition 5. We also have 5.

Using formulas 5. Using the formulas 5. Since Hr,s is an irreducible G-module, by 5. According to Proposition 5. Thus when s is even resp.

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Let X be the complex quadric Q4. Moreover by 1. The following lemma is a consequence of Proposition 5. Vanishing results for symmetric forms Proposition 5. Vanishing results for symmetric forms This section is mainly devoted to results concerning the sections of the vector bundle L and to the proofs of the following two results: Proposition 5.

Theorem 5.