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

JACQUES GASQUI
HUBERT GOLDSCHMIDT
Pages: 376
Stable URL: http://www.jstor.org/stable/j.ctt7t8th

1. Front Matter (pp. i-iv)
3. INTRODUCTION (pp. ix-xviii)

This monograph is motivated by a fundamental rigidity problem in Riemannian geometry: determine whether the metric of a given Riemannian symmetric space of compact type can be characterized by means of the spectrum of its Laplacian. An infinitesimal isospectral deformation of the metric of such a symmetric space belongs to the kernel of a certain Radon transform defined in terms of integration over the flat totally geodesic tori of dimension equal to the rank of the space. Here we study an infinitesimal version of this spectral rigidity problem: determine all the symmetric spaces of compact type for which this Radon...

4. CHAPTER I SYMMETRIC SPACES AND EINSTEIN MANIFOLDS (pp. 1-31)

LetXbe a differentiable manifold of dimensionn, whose tangent and cotangent bundles we denote by$T\; = \;{T_X}$and$T{\kern 1pt} *\; = \;T_X^*$, respectively. Let${C^\infty }(X)$be the space of complex-valued functions onX. By${ \otimes ^k}E,\;{S^l}E,\;{ \wedge ^j}{\kern 1pt} E$, we shall mean thek-th tensor product, thel-th symmetric product and thej-th exterior product of a vector bundleEoverX, respectively. We shall identify${S^k}T{\kern 1pt} *$and${ \wedge ^k}T{\kern 1pt} *$with sub-bundles of${ \otimes ^k}T{\kern 1pt} *$by means of the injective mappings

${S^k}T{\kern 1pt} *\; \to \;{ \otimes ^k}T{\kern 1pt} *,\quad \quad { \wedge ^k}{\kern 1pt} T{\kern 1pt} *\; \to \;{ \otimes ^k}{\kern 1pt} T{\kern 1pt} *$,

sending the symmetric product${\beta _1} \cdot \; \ldots \cdot \;{\beta _k}$into

$\sum\limits_{\sigma \in \,{\mathfrak{S}_k}} {{\beta _{^{\sigma \,(1)}}} \otimes \; \cdots \otimes \;{\beta _{\sigma \,(k)}}}$

and the exterior product${\beta _1}\, \wedge \; \cdots \; \wedge \;{\beta _k}$into

$\sum\limits_{\sigma \in \,{\mathfrak{S}_k}} {{\text{sgn}}\;\sigma \; \cdot \;{\beta _{\sigma \,(1)}}} \otimes \cdots \otimes \,{\beta _{\sigma \,(k)}},$

where${\beta _1},\, \ldots ,\;{\beta _k}\; \in \;T{\kern 1pt} *$and${\mathfrak{S}_k}$is the group of permutations of$\{ 1,\; \ldots ,\;k\}$and...

5. CHAPTER II RADON TRANSFORMS ON SYMMETRIC SPACES (pp. 32-74)

In this chapter, we introduce the Radon transforms for functions and symmetric forms on a symmetric space$(X,g)$of compact type, namely the X-ray transform and the maximal flat transform. In §2, we present results concerning harmonic analysis on homogeneous spaces and use them to study these Radon transforms in §5 and to describe properties of certain spaces of symmetric forms in §7. The notions of rigidity in the sense of Guillemin and of infinitesimal rigidity of the spaceXare introduced in §3; in this section, we also state the fundamental result of Guillemin [35] concerning isospectral deformations of...

6. CHAPTER III SYMMETRIC SPACES OF RANK ONE (pp. 75-113)

Let$(X,g)$be a flat Riemannian manifold of dimensionn. We first suppose thatXis the circle${S^1}$of lengthLendowed with the Riemannian metric$g\; = \;dt\; \otimes \;dt$, wheretis the canonical coordinate of${S^1}$defined moduloL. It is easily seen that this spaceXis infinitesimally rigid and that a 1-form onXsatisfies the zero-energy condition if and only if it is exact.

In this section, we henceforth suppose that$n\; \geqslant \;2$. We recall that$\tilde B\; = \;\{ 0\}$, that the operator${D_1}$is equal to${D_g}$, and that the sequence (1.50) is exact. Lethbe a section of...

7. CHAPTER IV THE REAL GRASSMANNIANS (pp. 114-133)

Let$m\; \geqslant \;1,\;n\; \geqslant \;0$be given integers and letFbe a real vector space of dimension$m + \;n$endowed with a positive definite scalar product. LetXbe the real Grassmannian$\tilde G_m^\mathbb{R}(F)$of all orientedm-planes inF.

Let$V\; = \;{V_X}$be the canonical vector bundle (of rankm) overXwhose fiber at$x\; \in \;X$is the subspace ofFdetermined by the orientedm-planex. We denote by$W\; = \;{W_X}$the vector bundle of ranknoverXwhose fiber at$x \in X$is the orthogonal complement${W_x}$of${V_x}$inF. Then we have a natural isomorphism of vector bundles

(4.1)$V{\kern 1pt} *\; \otimes \;W\; \to \;T$

over...

8. CHAPTER V THE COMPLEX QUADRIC (pp. 134-192)

This chapter is devoted to the complex quadric which plays a central role in the rigidity problems. In §§2 and 3, we describe the differential geometry of the quadric${Q_n}$viewed as a complex hypersurface of the complex projective space$\mathbb{C}{\mathbb{P}^{n\, + \,1}}$. We show that${Q_n}$is a Hermitian symmetric space and a homogeneous space of the group$SO(n\; + \;2)$. The involutions of the tangent spaces of${Q_n}$, which arise from the second fundamental form of the quadric, allow us to introduce various objects and vector bundles on${Q_n}$. In particular, we decompose the bundle of symmetric 2-forms on${Q_n}$into irreducible...

9. CHAPTER VI THE RIGIDITY OF THE COMPLEX QUADRIC (pp. 193-243)

In §2, we describe an explicit totally geodesic flat torus of the complex quadric${Q_n}$, with$n\; \geqslant \;3$, viewed as a complex hypersurface of projective space. In §3, we introduce certain symmetric 2-forms on the quadric; later, in §7 we shall see that they provide us with explicit bases for the highest weight subspaces of the isotypic components of the$SO(5)$-module of complex symmetric 2-forms on the three-dimensional quadric${Q_3}$. In §§4 and 5, we compute integrals over closed geodesics in order to prove that linear combinations of the symmetric 2-forms of §3 satisfying the zero-energy condition must verify certain relations....

10. CHAPTER VII THE RIGIDITY OF THE REAL GRASSMANNIANS (pp. 244-256)

Let$m\; \geqslant \;2$and$n\; \geqslant \;3$be given integers. We consider the real Grassmannians$X\; = \;\tilde G_{m,\,n}^\mathbb{R}$and$Y\; = \;G_{m,\,n}^\mathbb{R}$, endowed with the Riemannian metricsgandgydefined in §1, Chapter IV, and the natural Riemannian submersion$\varpi :\;X\; \to \;Y$. As in §1, Chapter IV, we view these Grassmannians as irreducible symmetric spaces and as homogeneous spaces of the group$G\; = \;SO(m\; + \;n)$. We identify the tangent bundleTofXwith the vector bundle$V \otimes \;W$. We shall also consider the Kähler metric${\tilde g}$on the complex quadric${Q_n}$defined in §2, Chapter V and denoted there byg.

Letxbe a point ofX. Let${\mathcal{F}_x}$...

11. CHAPTER VIII THE COMPLEX GRASSMANNIANS (pp. 257-307)

This chapter is devoted to the geometry of the complex Grassmannians. In §2, we study the complex Grassmannian$G_{m,\,n}^\mathbb{C}$, of complexm-planes in${\mathbb{C}^{m\, + \,n}}$, with$m,n\; \geqslant \;2$, and show that it is a Hermitian symmetric space and a homogeneous space of the group$SU(m\; + \;n)$; we also consider the Grassmannian$\bar G_{n,\,n}^\mathbb{C}$, which is the adjoint space of$G_{n,\,n}^\mathbb{C}$. We introduce certain vector bundles over$G_{m,\,n}^\mathbb{C}$and use them to decompose the bundle of symmetric 2-forms on$G_{m,\,n}^\mathbb{C}$into irreducible$SU(m\; + \;n)$-invariant sub-bundles. We then determine the highest weights of the fibers of these vector bundles in §3. We define certain complex-valued functions on...

12. CHAPTER IX THE RIGIDITY OF THE COMPLEX GRASSMANNIANS (pp. 308-328)

Let$m,n\; \geqslant \;2$be given integers. LetXbe the complex Grassmannian$G_{m,\,n}^\mathbb{C}$endowed with its Kähler metricg. As in §2, Chapter VIII, we view this Grassmannian as an irreducible symmetric space and as a homogeneous space of the group$G\; = \;SU(m\; + \;n)$, and we identify the tangent bundleTofXwith the complex vector bundle$V{\kern 1pt} *\;{ \otimes _\mathbb{C}}W$.

Letxbe a point ofX. Let$\mathcal{F}_x^1$be the family of all closed connected totally geodesic surfaces ofXpassing throughxof the form${\text{Ex}}{{\text{p}}_x}F$, whereFis generated (over$\mathbb{R}$) by the vectors$\{ {\alpha _1}\; \otimes \;{w_1},\;{\alpha _2}\; \otimes \;{w_2}\}$, where$\{ {\alpha _1},\;{\alpha _2}\}$is an orthonormal set...

13. CHAPTER X PRODUCTS OF SYMMETRIC SPACES (pp. 329-356)

LetYandZbe two manifolds; we consider the product$X\; = \;Y \times Z$and the natural projections${\text{p}}{{\text{r}}_Y}$and${\text{p}}{{\text{r}}_Z}$ofXontoYandZ, respectively. If$\theta$is a section of${ \otimes ^p}T_Y^*$overY(resp. of${ \otimes ^p}T_Z^*$overZ), we shall also denote by$\theta$the section${\text{pr}}_Y^*\theta$(resp. the section${\text{pr}}_Z^*\theta$) of${ \otimes ^p}{\kern 1pt} T{\kern 1pt} *$overX; a vector field$\xi$onY(resp. onZ) induces a vector field on the productX, which we shall also denote by$\xi$. If${\theta _1}$is a symmetricp-form onYand${\theta _2}$is a symmetricq-form onZ, we shall consider...

14. REFERENCES (pp. 357-362)
15. INDEX (pp. 363-366)