Tu L.W. Differential Geometry: Connections, Curvature, and Characteristic Classes

Springer International Publishing AG, 2017. — 358 p. — (Graduate Texts in Mathematics 275) — ISBN: 9783319550824.Differential geometry has a long and glorious history. As its name implies, it is the study of geometry using differential calculus, and as such, it dates back to Newton and Leibniz in the seventeenth century. But it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein’s general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. It has even found applications to group theory as in Gromov’s work and to probability theory as in Diaconis’s work. It is not too far-fetched to argue that differential geometry should be in every mathematician’s arsenal.
The basic objects in differential geometry are manifolds endowed with a metric, which is essentially a way of measuring the length of vectors. A metric gives rise to notions of distance, angle, area, volume, curvature, straightness, and geodesics. It is the presence of a metric that distinguishes geometry from topology. However, another concept that might contest the primacy of a metric in differential geometry is that of a connection. A connection in a vector bundle may be thought of as a way of differentiating sections of the vector bundle. A metric determines a unique connection called a Riemannian connection with certain desirable properties. While a connection is not as intuitive as a metric, it already gives rise to curvature and geodesics. With this, the connection can also lay claim to be a fundamental notion of differential geometry.Curvature and Vector Fields
Riemannian Manifolds
Curves
Surfaces in Space
Directional Derivatives in Euclidean Space
The Shape Operator
Affine Connections
Vector Bundles
Gauss’s Theorema Egregium
Generalizations to Hypersurfaces in Rn+1
Curvature and Differential Forms
Connections on a Vector Bundle
Connection, Curvature, and Torsion Forms
The Theorema Egregium Using Forms
Geodesics
More on Affine Connections
Geodesics
Exponential Maps
Distance and Volume
The Gauss–Bonnet Theorem
Tools from Algebra and Topology
The Tensor Product and the Dual Module
The Exterior Power
Operations on Vector Bundles
Vector-Valued Forms
Vector Bundles and Characteristic Classes
Connections and Curvature Again
Characteristic Classes
Pontrjagin Classes
The Euler Class and Chern Classes
Some Applications of Characteristic Classes
Principal Bundles and Characteristic Classes
Principal Bundles
Connections on a Principal Bundle
Horizontal Distributions on a Frame Bundle
Curvature on a Principal Bundle
Covariant Derivative on a Principal Bundle
Characteristic Classes of Principal Bundles
Appendixes
Manifolds
Invariant Polynomials