Xll

Introduction

contributed to their development were N. I. Lobachevsky, J. Bolyai,

C. F. Gauss, and E. Beltrami.

A detailed theory of surfaces in three-dimensional space was de-

veloped by C. F. Gauss. His main result was the Theorema Egregium,

which states that the curvature of a surface is an "intrinsic" property

of the surface. This means it can be measured and "felt" by someone

who is on the surface, rather than only by observing the surface from

outside.

However, the fundamental question "What is geometry?" still

remained. There are two directions of development after Gauss. The

first, is related to the work of B. Riemann, who conceived a framework

of generalizing the theory of surfaces of Gauss, from two to several

dimensions. The new objects are called Riemannian manifolds, where

a notion of curvature is defined, and is allowed to vary from point to

point, as in the case of a surface. Riemann brought the power of

calculus into geometry in an emphatic way as he introduced metrics

on the spaces of tangent vectors. The result is today called differential

geometry.

The other direction is the one developed by F. Klein, who used

the notion of a transformation group to define geometry. According to

Klein, the objects of study in geometry are the invariant properties

of geometrical figures under the actions of specific transformation

groups. Hence, the consideration of different transformation groups

leads to different kinds of geometry, such as Euclidean geometry, affine

geometry, or projective geometry. For example, Euclidean geometry

is the study of those properties of the plane that remain invariant

under the group of rigid motions of the plane (the Euclidean group).

The groups that were available at that time, and which Klein used

to determine various geometries, were developed by the Norwegian

mathematician Sophus Lie, and are now called Lie groups.

This brings us to the other terms of the title of this book, namely

"Lie groups" and "homogeneous spaces". The theory of Lie has its

roots in the study of symmetries of systems of differential equations,

and the integration techniques for them. At that time, Lie had called

these symmetries "continuous groups". In fact, his main goal was

to develop an analogue of Galois theory for differential equations.