Transformation Groups in Differential Geometry

verfasst von: Shoshichi Kobayashi

Verlag: Springer Berlin Heidelberg

Buchreihe : Ergebnisse der Mathematik und ihrer Grenzgebiete

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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

Frontmatter

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

Titel: Transformation Groups in Differential Geometry
verfasst von: Shoshichi Kobayashi
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-61981-6
Print ISBN: 978-3-540-58659-3
DOI: https://doi.org/10.1007/978-3-642-61981-6

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

I. Automorphisms of G-Structures

II. Isometries of Riemannian Manifolds

III. Automorphisms of Complex Manifolds

IV. Affine, Conformal and Projective Transformations

Backmatter