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

Gasqui, Jacques; Goldschmidt, Hubert

Softcover

ISBN 10: 069111899X ISBN 13: 9780691118994

Verlag: Princeton University Press, 2004

Inhaltsangabe

This book provides the first unified examination of the relationship between Radon transforms on symmetric spaces of compact type and the infinitesimal versions of two fundamental rigidity problems in Riemannian geometry. Its primary focus is the spectral rigidity problem: Can the metric of a given Riemannian symmetric space of compact type be characterized by means of the spectrum of its Laplacian? It also addresses a question rooted in the Blaschke problem: Is a Riemannian metric on a projective space whose geodesics are all closed and of the same length isometric to the canonical metric?


The authors comprehensively treat the results concerning Radon transforms and the infinitesimal versions of these two problems. Their main result implies that most Grassmannians are spectrally rigid to the first order. This is particularly important, for there are still few isospectrality results for positively curved spaces and these are the first such results for symmetric spaces of compact type of rank >1. The authors exploit the theory of overdetermined partial differential equations and harmonic analysis on symmetric spaces to provide criteria for infinitesimal rigidity that apply to a large class of spaces.


A substantial amount of basic material about Riemannian geometry, symmetric spaces, and Radon transforms is included in a clear and elegant presentation that will be useful to researchers and advanced students in differential geometry.

Über diesen Titel

Über die Autorin bzw. den Autor

Jacques Gasqui is Professor of Mathematics at Institut Fourier, Universite de Grenoble I. Hubert Goldschmidt is Visiting Professor of Mathematics at Columbia University and Professeur des Universites in France.

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Radon Transforms and the Rigidity of the Grassmannians

By JACQUES GASQUI HUBERT GOLDSCHMIDT

PRINCETON UNIVERSITY PRESS

Copyright © 2004 Princeton University Press
All right reserved.
ISBN: 978-0-691-11899-4

Contents

Introduction............................................................................................ix1. Riemannian manifolds.................................................................................12. Einstein manifolds...................................................................................153. Symmetric spaces.....................................................................................194. Complex manifolds....................................................................................271. Outline..............................................................................................322. Homogeneous vector bundles and harmonic analysis.....................................................323. The Guillemin and zero-energy conditions.............................................................364. Radon transforms.....................................................................................415. Radon transforms and harmonic analysis...............................................................506. Lie algebras.........................................................................................587. Irreducible symmetric spaces.........................................................................598. Criteria for the rigidity of an irreducible symmetric space..........................................681. Flat tori............................................................................................752. The projective spaces................................................................................833. The real projective space............................................................................894. The complex projective space.........................................................................945. The rigidity of the complex projective space.........................................................1046. The other projective spaces..........................................................................1121. The real Grassmannians...............................................................................1142. The Guillemin condition on the real Grassmannians....................................................1261. Outline..............................................................................................1342. The complex quadric viewed as a symmetric space......................................................1343. The complex quadric viewed as a complex hypersurface.................................................1384. Local Khler geometry of the complex quadric.........................................................1465. The complex quadric and the real Grassmannians.......................................................1526. Totally geodesic surfaces and the infinitesimal orbit of the curvature...............................1597. Multiplicities.......................................................................................1708. Vanishing results for symmetric forms................................................................1859. The complex quadric of dimension two.................................................................1901. Outline..............................................................................................1932. Total geodesic flat tori of the complex quadric......................................................1943. Symmetric forms on the complex quadric...............................................................1994. Computing integrals of symmetric forms...............................................................2045. Computing integrals of odd symmetric forms...........................................................2096. Bounds for the dimensions of spaces of symmetric forms...............................................2187. The complex quadric of dimension three...............................................................2238. The rigidity of the complex quadric..................................................................2299. Other proofs of the infinitesimal rigidity of the quadric............................................23210. The complex quadric of dimension four...............................................................23411. Forms of degree one.................................................................................2371. The rigidity of the real Grassmannians...............................................................2442. The real Grassmannians [[bar.G].sup.R.sub.n,n].......................................................2491. Outline..............................................................................................2572. The complex Grassmannians............................................................................2583. Highest weights of irreducible modules associated with the complex Grassmannians.....................2704. Functions and forms on the complex Grassmannians.....................................................2745. The complex Grassmannians of rank two................................................................2826. The Guillemin condition on the complex Grassmannians.................................................2877. Integrals of forms on the complex Grassmannians......................................................2938. Relations among forms on the complex Grassmannians...................................................3009. The complex Grassmannians [[bar.G].sup.C.sub.n,n]....................................................3031. The rigidity of the complex Grassmannians............................................................3082. On the rigidity of the complex Grassmannians [[bar.G].sup.C.sub.n,n].................................3133. The rigidity of the quaternionic Grassmannians.......................................................3231. Guillemin rigidity and products of symmetric spaces..................................................3292. Conformally flat symmetric spaces....................................................................3343. Infinitesimal rigidity of products of symmetric spaces...............................................3384. The infinitesimal rigidity of [[bar.G].sup.R.sub.2,2]................................................340References..............................................................................................357Index...................................................................................................363

Chapter One

SYMMETRIC SPACES AND EINSTEIN MANIFOLDS

1. Riemannian manifolds

Let X be a differentiable manifold of dimension n, whose tangent and cotangent bundles we denote by T = [T.sub.X] and [T.sup.*] = [T.sup.*.sub.X], respectively. Let [ITLITL.sup.[infinity]](X) be the space of complex-valued functions on X. By [[cross product].sup.k] E, [S.sup.l] E, [[LAMBDA].sup.j]E, we shall mean the k-th tensor product, the l-th symmetric product and the j-th exterior product of a vector bundle E over X, respectively. We shall identify [S.sup.k][T.sup.*] and [[LAMBDA].sup.k][T.sup.*] with sub-bundles of [[cross product].sup.k][T.sup.*] by means of the injective mappings

[S.sup.k][T.sup.*] -> [[cross product].sup.k][T.sup.*], [[LAMBDA].sup.k][T.sup.*] -> [[cross product].sup.k][T.sup.*]

sending the symmetric product [.sub.1] ... [.sub.k] into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the exterior product [.sub.1] [and] [and] [.sub.k] into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [.sub.1], ..., [.sub.k] [member of] [T.sup.*] and [[??].sub.k] is the group of permutations of {1, ..., k} and sgn [sigma] is the signature of the element [sigma] of [[??].sub.k]. If [alpha], [member of] [T.sup.*], the symmetric product [alpha] is identified with the element [alpha] [cross product] + [cross product] [alpha] of [[cross product].sup.2][T.sup.*]. If [xi] [member of] T, h [member of] [S.sup.2][T.sup.*], let [xi] [??] h be the element of [T.sup.*] defined by

([xi] [??] h)([eta]) = h([xi], [eta]),

for [eta] [member of] T. If h [member of] [S.sup.2][T.sup.*], we denote by

[h.sup.b] : T -> [T.sup.*]

the mapping sending [xi] [member of] T into [xi] [??] h. If h is non-degenerate, then [h.sup.b] is an isomorphism, whose inverse will be denoted by [h.sup.#].

Let E be a vector bundle over X; we denote by [E.sub.C] its complexification, by E the sheaf of sections of E over X and by ITLITL.sup.[infinity](E) the space of global sections of E over X. We write [S.sup.l] [E.sub.C] and [[LAMBDA].sup.j][E.sup.C] for the complexifications of [S.sub.l] E and [[LAMBDA].sup.j][E. We consider the vector bundle [J.sub.k](E) of k-jets of sections of E, whose fiber at x [member of] X is the quotient of the space [ITLITL.sup.[infinity]](E) by its subspace consisting of the sections of E which vanish to order k + 1 at x. If s is a section of E over X, the k-jet [j.sub.k](s)(x) of s at x [member of] X is the equivalence class of s in [J.sub.k][(E).sub.x]. The mapping x -> [j.sub.k](s)(x) is a section [j.sub.k](s) of [J.sub.k](E) over X. We denote by [[pi].sub.k] : [J.sub.k+l](E) -> [J.sub.k](E) the natural projection sending [j.sub.k+l](s)(x) into [j.sub.k](s)(x), for x [member of] X. We shall identify [J.sub.0](E) with E and we set [J.sub.k](E) = 0, for k < 0. The morphism of vector bundles

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

determined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [f.sub.1], ..., [f.sub.k] are real-valued functions on X vanishing at x [member of] X and s is a section of E over X, is well-defined since the function [[PI].sup.k.sub.i=1][f.sub.i] vanishes to order k - 1 at x. One easily verifies that the sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is exact, for k [greater than or equal to] 0.

Let E and F be vector bundles over X. If D : E -> F is a differential operator of order k, there exists a unique morphism of vector bundles

p(D) : [J.sub.k](E) -> F

such that

Ds = p(D)[j.sub.k](s),

for all s [member of] E. The symbol of D is the morphism of vector bundles

[sigma](D) : [S.sup.k]]T.sup.*] [cross product] E -> F

equal to p(D) o [epsilon]. If x [member of] X and [alpha] [member of] [T.sup.*.sub.x], let

[[sigma].sub.[alpha]](D) : [E.sub.x] -> [F.sub.x]

be the linear mapping defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where u [member of] [E.sub.x] and [[alpha].sub.k] denotes the k-th symmetric product of [alpha]. We say that D is elliptic if, for all x [member of] X and [alpha] [member of] [T.sup.*.sub.x], the mapping [[sigma].sub.[alpha]](D) is injective. If D is elliptic and X is compact, then it is well-known that the kernel of D : [ITLITL.sup.[infinity]](E) -> [ITLITL.sup.[infinity]](F) is finite-dimensional.

If [xi] is a vector field on X and is a section of [[cross product].sup.k][T.sup.*] over X, we denote by [L.sub.[xi]] the Lie derivative of along [xi]. For x [member of] X, let [rho] denote the representation of the Lie algebra [([T.sup.*][cross product]T).sub.x] on [T.sub.x] and also the representation induced by [rho] on [[cross product].sup.k][T.sup.*.sub.x]. If [xi] is an element of [T.sub.x] satisfying [xi](x) = 0, and if u is the unique element of [([T.sup.*] [cross product] T).sub.x] determined by the relation [epsilon](u) = [j.sub.1](xi)(x), then we have

[rho](u)]eta](x) = -[[xi], [eta]](x),

for all [eta] [member of] [T.sub.x], and

[rho](u)(x) = -(L]epsilon])(x), (1.1)

for [member of] [[cross product].sup.k][T.sup.*.sub.x].

We endow X with a Riemannian metric g and we associate various objects to the Riemannian manifold (X, g). The mappings [g.sup.b] : T -> [T.sup.*], [g.sup.#]: [T.sup.*] -> T are the isomorphisms determined by the metric g; we shall sometimes write [[xi].sup.b] = [g.sup.b] ([xi]) and [[alpha].sup.#] = [g.sup.#]([alpha]), for [xi] [member of] T and [alpha] [member of] [T.sup.*]. The metric g induces scalar products on the vector bundle [[cross product].sup.p][T.sup.*] [cross product] [[cross product].sup.q]T and its sub-bundles. We denote by dX the Riemannian measure of the Riemannian manifold (X, g). If X is compact, the volume Vol (X, g) of (X, g) is equal to the integral [[integral].sub.X]dX.

Let E and F be vector bundles over X endowed with scalar products and D : E -> F be a differential operator of order k. We consider the scalar products on [ITLITL.sup.[infinity]](E) and [ITLITL.sup.[infinity]](F), defined in terms of these scalar products on E and F and the Riemannian measure of X, and the formal adjoint [D.sup.*]: F -> E of D, which is a differential operator of order k. If D is elliptic and X is compact, then [DC.sup.[infinity]](E) is a closed subspace of [ITLITL.sup.[infinity]](F) and we have the orthogonal decomposition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Let B = [B.sub.X] be the sub-bundle of [[LAMBDA].sup.2][T.sup.*] [cross product] [[LAMBDA].sup.2][T.sup.*] consisting of those tensors u [member of] [[LAMBDA].sup.2][T.sup.*] which satisfy the first Bianchi identity

u([[xi].sub.1], [[xi].sub.2], [[xi].sub.3], [[xi].sub.4]) + u([[xi].sub.2], [[xi].sub.3], [[xi].sub.1], [[xi].sub.4]) + u([[xi].sub.3], [[xi].sub.1], [[xi].sub.2], [[xi].sub.4]) = 0,

for all [xi].sub.1], [[xi].sub.2], [[xi].sub.3], [[xi].sub.4] [member of] T. It is easily seen that, if an element u of B satisfies the relation

u([xi].sub.1], [[xi].sub.2], [[xi].sub.1], [[xi].sub.2]) = 0,

for all [xi].sub.1], [[xi].sub.2] [member of] T, then u vanishes. We consider the morphism of vector bundles

[tau]B : [S.sup.2][T.sup.*] [cross product] [S.sup.2][T.sup.*] -> B

defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all u [member of] [S.sup.2][T.sup.*] and [xi].sub.1], [[xi].sub.2], [[xi].sub.3], [[xi].sub.4] [member of] T; it is well-known that this morphism is an epimorphism (see Lemma 3.1 of [13]). Let

[sigma] : [T.sup.*] [cross product] B -> [[LAMBDA].sup.3][T.sup.*] [cross product] [[LAMBDA].sup.*][T.sup.*]

be the restriction of the morphism of vector bundles

[T.sup.*] [cross product] [[LAMBDA].sup.*][T.sup.*] [cross product] [[LAMBDA].sup.*][T.sup.*] -> [cross product] [[LAMBDA].sup.*][T.sup.*] [cross product] [[LAMBDA].sup.*][T.sup.*],

which sends [alpha] [cross product] [[theta].sub.1] [cross product] [[theta].sub.2] into ([alpha] [and] [[theta].sub.1]) [cross product] [[theta].sub.2], for [alpha] [member of] [T.sup.*], [[theta].sub.1], [[theta].sub.2] [member of] [[LAMBDA].sup.2][T.sup.*]. The kernel H of this morphism [sigma] is equal to the sub-bundle of [T.sup.*] [cross product] B consisting of those tensors v [member of] [T.sup.*] B which satisfy the relation

v([[xi].sub.1], [[xi].sub.2], [[xi].sub.3], [[xi].sub.4], [[xi].sub.5]) + v([[xi].sub.2], [[xi].sub.3], [[xi].sub.1], [[xi].sub.4], [[xi].sub.5]) + v([[xi].sub.3], [[xi].sub.1], [[xi].sub.2], [[xi].sub.4], [[xi].sub.5]) = 0,

for all [[xi].sub.1], [[xi].sub.2], [[xi].sub.3], [[xi].sub.4], [[xi].sub.5] [member of] T.

Let

Tr = [Tr.sub.g] = [Tr.sub.X] : [S.sup.2][T.sup.*] -> R, Tr = [Tr.sub.g] : [[LAMBDA].sup.2][T.sup.*] -> [[cross product].sup.2][T.sup.*]

be the trace mappings defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [xi], [eta] [member of] [T.sub.x], where x [member of] X and {[t.sub.1], ..., [t.sub.n]} is an orthonormal basis of [T.sub.x]. It is easily seen that

Tr B [subset] [S.sup.2][T.sup.*].

We denote by [S.sup.2.sub.0][T.sup.*] the sub-bundle of [S.sup.2][T.sup.*] equal to the kernel of the trace mapping Tr : [S.sup.2][T.sup.*] -> R and by [B.sup.0] the sub-bundle of B equal to the kernel of the trace mapping Tr : B -> [S.sup.2][T.sup.*].

We consider the morphism of vector bundles

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for h [member of] [S.sup.2][T.sup.*]. If h [member of] [S.sup.2][T.sup.*], we easily verify that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

and so we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When n [greater than or equal to] 3, from the preceding formulas we infer that the morphism [[??].sub.B] is injective, that the morphism Tr : B -> [S.sup.2][T.sup.*] is surjective and that [B.sup.0] is the orthogonal complement of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in B.

We introduce various differential operators and objects associated with the Riemannian manifold (X, g). First, let [nabla] = [[nabla].sup.g] be the Levi-Civita connection of (X, g). If f is a real-valued function on X, we denote by Hess f = [nabla]df the Hessian of f. If [d.sup.*] : [[LAMBDA].sup.j] [T.sup.*] -> [[LAMBDA].sup.j-1][T.sup.*] is the formal adjoint of the exterior differential operator d : [[LAMBDA].sup.j-1][T.sup.*] -> [[LAMBDA].sup.j][T.sup.*], we consider the de Rham Laplacian [DELTA] = [dd.sup.*] + [d.sup.*]d acting on [[LAMBDA].sup.j][T.sup.*]. The Laplacian [DELTA] = [[DELTA].sup.g] = [[DELTA].sup.X] acting on [ITLITL.sup.[infinity]](X) is also determined by the relation [DELTA]f = -Tr Hess f, for f [member of] [ITLITL.sup.[infinity]](X). The spectrum Spec(X, g) of the metric g on X is the sequence of eigenvalues (counted with multiplicities)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of the Laplacian [[DELTA].sub.g] acting on the space [ITLITL.sup.[infinity]](X).

The Killing operator

[D.sub.0] = [D.sub.0,X] : T -> [S.sup.2][T.sup.*]

of (X, g), which sends [xi] [member of] T into [L.sub.[xi]]g, and the symmetrized covariant derivative

[D.sup.1] : [T.sup.*] -> [S.sup.2][T.sup.*],

defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for [theta] [member of] [T.sup.*], [xi], [eta] [member of] T, are related by the formula

1/2 [D.sub.0][xi] = [D.sup.1][g.sup.b](xi), (1.4)

for [xi] [member of] T . By (1.4), the conformal Killing operator

[D.sup.c.su.0] : T -> [S.sup.2.sub.0][T.sup.*]

of (X, g) is determined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for [xi] [member of] T. The Killing (resp. conformal Killing) vector fields of (X, g) are the solutions [xi] [member of] [ITLITL.sup.[infinity]](T) of the equation [D.sub.0][xi] = 0 (resp. [D.sup.c.sub.0][xi] = 0). A vector field [xi] on X is a conformal Killing vector field if and only if there is a real-valued function f such that [D.sub.0][xi] = fg. According to (1.4), a Killing vector field [xi] on X satisfies the relation [d.sup.*][g.sup.b]([xi]) = 0, and a real-valued function f on X satisfies the relation

[D.sub.0][(df).sup.#] = 2 Hess f. (1.5)

If [theta] is a local isometry of X defined on an open subset U of X and [xi] is a vector field on U, according to (1.4) we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

(Continues...)

Excerpted from Radon Transforms and the Rigidity of the Grassmanniansby JACQUES GASQUI HUBERT GOLDSCHMIDT Copyright © 2004 by Princeton University Press. Excerpted by permission.
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Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2004
• ISBN 10: 069111899X
• ISBN 13: 9780691118994
• Einband: Tapa blanda
• Sprache: Englisch
• Anzahl der Seiten: 386
• Kontakt zum Hersteller: Nicht verfügbar