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## Über dieses Buch

Given a mathematical structure, one of the basic associated mathematical objects is its automorphism group. The object of this book is to give a biased account of automorphism groups of differential geometric struc­ tures. All geometric structures are not created equal; some are creations of ~ods while others are products of lesser human minds. Amongst the former, Riemannian and complex structures stand out for their beauty and wealth. A major portion of this book is therefore devoted to these two structures. Chapter I describes a general theory of automorphisms of geometric structures with emphasis on the question of when the automorphism group can be given a Lie group structure. Basic theorems in this regard are presented in §§ 3, 4 and 5. The concept of G-structure or that of pseudo-group structure enables us to treat most of the interesting geo­ metric structures in a unified manner. In § 8, we sketch the relationship between the two concepts. Chapter I is so arranged that the reader who is primarily interested in Riemannian, complex, conformal and projective structures can skip §§ 5, 6, 7 and 8. This chapter is partly based on lec­ tures I gave in Tokyo and Berkeley in 1965.

## Inhaltsverzeichnis

### I. Automorphisms of G-Structures

Abstract
Let M be a differentiable manifold of dimension n and L(M) the bundle of linear frames over M. Then L(M) is a principal fibre bundle over M with group GL(n; R). Let G be a Lie subgroup of GL(n; R). By a G-structure on M we shall mean a differentiable subbundle P of L(M) with structure group G.
Shoshichi Kobayashi

### II. Isometries of Riemannian Manifolds

Abstract
The earliest and very general result on the group of isometries is perhaps the following theorem of van Danzig and van der Waerden [1] (see also Kobayashi-Nomizu [1, vol. 1; pp. 46–50] for a proof).
Shoshichi Kobayashi

### III. Automorphisms of Complex Manifolds

Abstract
Let M be a complex manifold and ℌ(M) the group of holomorphic transformations of M. In general, ℌ(M) can be infinite dimensional. For instance, ℌ(C n ) is not a Lie group if n≧2. To see this, consider transformations of C 2 of the form
$$\begin{array}{*{20}{c}} {z' = z} \\ {w' = w + f\left( z \right)} \\ \end{array} \left( {z,w} \right) \in {{C}^{2}}$$
where f(z) is an entire function in z, e. g., a polynomial of any degree in z. The fact that ℌ(C 2) contains these transformations shows that ℌ(C 2) cannot be finite dimensional. Similarly, for ℌ(C 2) with n≧2. On the other hand, ℌ(C) is the group of orientation preserving conformal transformations and, as we shall see later, it is a Lie group. The purpose of this section is to give conditions on M which imply that ℌ(M) is a Lie group.
Shoshichi Kobayashi

### IV. Affine, Conformal and Projective Transformations

Abstract
Let M be a manifold with an affine connection and L(M) be the bundle of linear frames over M. Let θ and ω denote the canonical form and the connection form on L(M) respectively. We recall (§ 1 of Chapter II) that a transformation f of M is said to be affine if the induced automorphism f ̄ of L(M) leaves ω invariant. We quote the following result established earlier (see Theorem 1.3 of Chapter II).
Shoshichi Kobayashi

### Backmatter

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