TABLE OF CONTENTS

INTRODUCTION

CHAPTER I

SYMMETRIC SPACES AND EINSTEIN MANIFOLDS

1. Riemannian manifolds

2. Einstein manifolds

3. Symmetric spaces

4. Complex manifolds

CHAPTER II

RADON TRANSFORMS ON SYMMETRIC SPACES

1. Outline

2. Homogeneous vector bundles and harmonic analysis.

3. The Guillemin and zero-energy conditions

4. Radon transforms

5. Radon transforms and harmonic analysis

6. Lie algebras

7. Irreducible symmetric spaces

8. Criteria for the rigidity of an irreducible symmetric space

CHAPTER III

SYMMETRIC SPACES OF RANK ONE

1. Flat tori

2. The projective spaces

3. The real projective space

4. The complex projective space

5. The rigidity of the complex projective space

6. The other projective spaces

CHAPTER IV

THE REAL GRASSMANNIANS

1. The real Grassmannians

2. The Guillemin condition on the real Grassmannians

CHAPTER V

THE COMPLEX QUADRIC

1. Outline

2. The complex quadric viewed as a symmetric space

3. The complex quadric viewed as a complex hypersurface

4. Local Kahler geometry of the complex quadric

5. The complex quadric and the real Grassmannians

6. Totally geodesic surfaces and the infinitesimal orbit of the curvature

7. Multiplicities

8. Vanishing results for symmetric forms

9. The complex quadric of dimension two

CHAPTER VI

THE RIGIDITY OF THE COMPLEX QUADRIC

1. Outline

2. Total geodesic flat tori of the complex quadric

3. Symmetric forms on the complex quadric

4. Computing integrals of symmetric forms

5. Computing integrals of odd symmetric forms

6. Bounds for the dimensions of spaces of symmetric forms

7. The complex quadric of dimension three

8. The rigidity of the complex quadric

9. Other proofs of the infinitesimal rigidity of the quadric

10. The complex quadric of dimension four

11. Forms of degree one

CHAPTER VII

THE RIGIDITY OF THE REAL GRASSMANNIANS

1. The rigidity of the real Grassmannians

2. The real Grassmannians

CHAPTER VIII

THE COMPLEX GRASSMANNIANS

1. Outline

2. The complex Grassmannians

3. Highest weights of irreducible modules associated with the complex Grassmannians

4. Functions and forms on the complex Grassmannians

5. The complex Grassmannians of rank two

6. The Guillemin condition on the complex Grassmannians

7. Integrals of forms on the complex Grassmannians

8. Relations among forms on the complex Grassmannians

9. The complex Grassmannians

CHAPTER IX

THE RIGIDITY OF THE COMPLEX GRASSMANNIANS

1. The rigidity of the complex Grassmannians

2. On the rigidity of the complex Grassmannians

3. The rigidity of the quaternionic Grassmannians

CHAPTER X

PRODUCTS OF SYMMETRIC SPACES

1. Guillemin rigidity and products of symmetric spaces

2. Conformally flat symmetric spaces

3. Infinitesimal rigidity of products of symmetric spaces

4. The infinitesimal rigidity of

REFERENCES