Differential geometry : connections, curvature, and characteristic classes
 Responsibility
 Loring W. Tu.
 Publication
 Cham, Switzerland : Springer, [2017]
 Physical description
 xvi, 346 pages : illustrations (some color), portraits (some color) ; 24 cm.
 Series
 Graduate texts in mathematics ; 275.
At the library
Science Library (Li and Ma)
Stacks
Call number  Status 

QA641 .T76 2017  Unknown 
More options
Description
Creators/Contributors
 Author/Creator
 Tu, Loring W., author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 335336) and index.
 Contents

 Preface
 Curvature and vector fields
 Riemannian manifolds
 Inner products on a vector space
 Representations of inner products by symmetric matrices
 Riemannian metrics
 Existence of a riemannian metric
 Problems
 Curves
 Regular curves
 Arc length parametrization
 Signed curvature of a plane curve
 Orientation and curvature
 Problems
 Surfaces in space
 Principal, mean, and gaussian curvatures
 Gauss's theorema egregium
 The gaussbonnet theorem
 Problems
 Directional derivatives in euclidean space
 Directional derivatives in euclidean space
 Other properties of the directional derivative
 Vector fields along a curve
 Vector fields along a submanifold
 Directional derivatives on a submanifold of R...
 Problems
 The shape operator
 Normal vector fields
 The shape operator
 Curvature and the shape operator
 The first and second fundamental forms
 The catenoid and the helicoid
 Problems
 Affine connections
 Affine connections
 Torsion and curvature
 The riemannian connection
 Orthogonal projection on a surface in R³
 The riemannian connection on a surface in R³
 Problems
 Vector bundles
 Definition of a vector bundle
 The vector space of sections
 Extending a local section to a global section
 Local operators
 Restriction of a local operator to an open subset
 Frames
 ...Linearity and bundle maps
 Multilinear maps over smooth functions
 Problems
 Gauss's theorema egregium
 The gauss and codazzimainardi equations
 A proof of the theorema egregium
 The gaussian curvature in terms of an arbitrary basis
 Problems
 Generalizations to hypersurfaces in R...
 The shape operator of a hypersurface
 The riemannian connection of a hypersurface
 The second fundamental form
 The gauss curvature and codazzimainardi equations
 Curvature and differential forms
 Connections on a vector bundle
 Connections on a vector bundle
 Existence of a connection on a vector bundle
 Curvature of a connection on a vector bundle
 Riemannian bundles
 Metric connections
 Restricting a connection to an open subset
 Connections at a point
 Problems
 Connection, curvature, and torsion forms
 Connection and curvature forms
 Connections on a framed open set
 The gramschmidt process
 Metric connection relative to an orthonormal frame
 Connections on the tangent bundle
 Problems
 The theorema egregium using forms
 The gauss curvature equation
 The theorema egregium
 Skewsymmetries of the curvature tensor
 Sectional curvature
 Poincaré halfplane
 Problems
 Geodesies
 More on affine connections
 Covariant differentiation along a curve
 Connectionpreserving diffeomorphisms
 Christoffel symbols
 Problems
 Geodesies
 The definition of a geodesic
 Reparametrization of a geodesic
 Existence of geodesies
 Geodesies in the poincaré halfplane
 Parallel translation
 Existence of parallel translation along a curve
 Parallel translation on a riemannian manifold
 Problems
 Exponential maps
 The exponential map of a connection
 The differential of the exponential map
 Normal coordinates
 Leftinvariant vector fields on a lie group
 Exponential map for a lie group
 Naturality of the exponential map for a lie group
 Adjoint representation
 Associativity of a biinvariant metric on a lie group
 Problems
 Addendum : the exponential map as a natural transformation
 Distance and volume
 Distance in a riemannian manifold
 Geodesic completeness
 Dual 1forms under a change of frame
 Volume form
 The volume form in local coordinates
 Problems
 The gaussbonnet theorem
 Geodesic curvature
 The angle function along a curve
 Signed geodesic curvature on an oriented surface
 Gaussbonnet formula for a polygon
 Triangles on a riemannian 2manifold
 Gaussbonnet theorem for a surface
 Gaussbonnet theorem for a hypersurface in R²...
 Problems
 Tools from algebra and topology
 The tensor product and the dual module
 Construction of the tensor product
 Universal mapping property for bilinear maps
 Characterization of the tensor product
 A basis for the tensor product
 The dual module
 Identities for the tensor product
 Functoriality of the tensor product
 Generalization to multilinear maps
 Associativity of the tensor product
 The tensor algebra
 Problems
 The exterior power
 The exterior algebra
 Properties of the wedge product
 Universal mapping property for alternating klinear maps
 A basis for ... V
 Nondegenerate pairings
 A nondegenerate pairing of .... with ... V
 A formula for the wedge product
 Problems
 Operations on vector bundles
 Vector subbundles
 Subbundle criterion
 Quotient bundles
 The pullback bundle
 Examples of the pullback bundle
 The direct sum of vector bundles
 Other operations on vector bundles
 Problems
 Vectorvalued forms
 Vectorvalued forms as sections of a vector bundle
 Products of vectorvalued forms
 Directional derivative of a vectorvalued function
 Exterior derivative of a vectorvalued form
 Differential forms with values in a lie algebra
 Pullback of vectorvalued forms
 Forms with values in a vector bundle
 Tensor fields on a manifold
 The tensor criterion
 Remark on signs concerning vectorvalued forms
 Problems
 Vector bundles and characteristic classes
 Connections and curvature again
 Connection and curvature matrices under a change of frame
 Bianchi identities
 The first bianchi identity in vector form
 Symmetry properties of the curvature tensor
 Covariant derivative of tensor fields
 The second bianchi identity in vector form
 Ricci curvature
 Scalar curvature
 Defining a connection using connection matrices
 Induced connection on a pullback bundle
 Problems
 Characteristic classes
 Invariant polynomials on g...(r, R)
 The chernweil homomorphism
 Characteristic forms are closed
 Differential forms depending on a real parameter
 Independence of characteristic classes of a connection
 Functorial definition of a characteristic class
 Naturality
 Problems
 Pontrjagin classes
 Vanishing of characteristic classes
 Pontrjagin classes
 The whitney product formula
 The euler class and chern classes
 Orientation on a vector bundle
 Characteristic classes of an oriented vector bundle
 The pfaffian of a skewsymmetric matrix
 The euler class
 Generalized gaussbonnet theorem
 Hermitian metrics
 Connections and curvature on a complex vector bundle
 Chern classes
 Problems
 Some applications of characteristic classes
 The generalized gaussbonnet theorem
 Characteristic numbers
 The cobordism problem
 The embedding problem
 The hirzebruch signature formula
 The riemannroch problem
 Principal bundles and characteristic classes
 Principal bundles
 Principal bundles
 The frame bundle of a vector bundle
 Fundamental vector fields of a right action
 Integral curves of a fundamental vector field
 Vertical subbundle of the tangent bundle TP
 Horizontal distributions on a principal bundle
 Problems
 Connections on a principal bundle
 Connections on a principal bundle
 Vertical and horizontal components of a tangent vector
 The horizontal distribution of an ehresmann connection
 Horizontal lift of a vector field to a principal bundle
 Lie bracket of a fundamental vector field
 Problems
 Horizontal distributions on a frame bundle
 Parallel translation in a vector bundle
 Horizontal vectors on a frame bundle
 Horizontal lift of a vector field to a frame bundle
 Pullback of a connection on a frame bundle under a section
 Curvature on a principal bundle
 Curvature form on a principal bundle
 Properties of the curvature form
 Problems
 Covariant derivative on a principal bundle
 The associated bundle
 The fiber of the associated bundle
 Tensorial forms on a principal bundle
 Covariant derivative
 A formula for the covariant derivative of a tensorial form
 Problems
 Characteristic classes of principal bundles
 Invariant polynomials on a lie algebra
 The chernweil homomorphism
 Problems
 Appendix
 Manifolds
 Manifolds and smooth maps
 Tangent vectors
 Vector fields
 Differential forms
 Exterior differentiation on a manifold
 Exterior differentiation on R³
 Pullback of differential forms
 Problems
 Invariant polynomials
 Polynomials versus polynomial functions
 Polynomial identities
 Invariant polynomials on g...(r, F)
 Invariant complex polynomials
 Lpolynormals, todd polynomials, and the chern character
 Invariant real polynomials
 Newton's identities
 Problems
 Hints and solutions to selected endofsection problems
 List of notations
 References
 Index.
 Publisher's Summary
 This text presents a graduatelevel introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the ChernWeil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the GaussBonnet theorem. Exercises throughout the book test the reader's understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text.Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more selfcontained, sections on algebraic constructions such as the tensor product and the exterior power are included.Differential geometry, as its name implies, is the study of geometry using differential calculus. 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. The field 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 farfetched to argue that differential geometry should be in every mathematician's arsenal.
(source: Nielsen Book Data)9783319550824 20170911
Subjects
 Subject
 Geometry, Differential.
Bibliographic information
 Publication date
 2017
 Title Variation
 Connections, curvature, and characteristic classes
 Series
 Graduate texts in mathematics, 00725285 ; 275
 Note
 Textbook for graduates.
 ISBN
 9783319550824 (hd.bd.)
 9783319550848 (online)
 3319550829
 3319550845 (eBook)