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
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
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.
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