Differential Geometry: Curves - Surfaces - Manifolds
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that arise in calculus. Here we learn about line and surface
integrals, divergence and curl, and the various forms of Stokes'
Theorem. If we are fortunate, we may encounter curvature and such
things as the Serret-Frenet formulas.
With just the basic tools
from multivariable calculus, plus a little knowledge of linear algebra,
it is possible to begin a much richer and rewarding study of
differential geometry, which is what is presented in this book. It
starts with an introduction to the classical differential geometry of
curves and surfaces in Euclidean space, then leads to an introduction
to the Riemannian geometry of more general manifolds, including a look
at Einstein spaces. An important bridge from the low-dimensional theory
to the general case is provided by a chapter on the intrinsic geometry
of surfaces.
The first half of the book, covering the geometry
of curves and surfaces, would be suitable for a one-semester
undergraduate course. The local and global theories of curves and
surfaces are presented, including detailed discussions of surfaces of
rotation, ruled surfaces, and minimal surfaces.
The second half
of the book, which could be used for a more advanced course, begins
with an introduction to differentiable manifolds, Riemannian
structures, and the curvature tensor. Two special topics are treated in
detail: spaces of constant curvature and Einstein spaces.
The
main goal of the book is to get started in a fairly elementary way,
then to guide the reader toward more sophisticated concepts and more
advanced topics. There are many examples and exercises to help along
the way. Numerous figures help the reader visualize key concepts and
examples, especially in lower dimensions. For the second edition, a
number of errors were corrected and some text and a number of figures
have been added.
