About the Book
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 230. Chapters: Manifold, Riemannian connection on a surface, Metric tensor, Affine connection, Cartan connection, Symmetric space, Holonomy, Covariance and contravariance of vectors, Ricci calculus, Information geometry, Shape of the Universe, Hopf fibration, Lie sphere geometry, Frenet-Serret formulas, Covariant derivative, List of nonlinear partial differential equations, Lie derivative, Envelope (mathematics), Jet (mathematics), Symmetry (physics), Connection form, Covariant transformation, Anti de Sitter space, Grassmannian, Clifford analysis, Tangent, Instanton, Darboux frame, Conformal geometry, Generalizations of the derivative, Torsion tensor, Generalized complex structure, Fisher information metric, Radius of curvature (applications), Minimal surface, Differential geometry of curves, Connection (mathematics), Principal bundle, Classification of manifolds, Moving frame, Petrov classification, Affine curvature, Heat kernel signature, Directional derivative, Tensor field, Sine-Gordon equation, Pedal curve, Gaussian curvature, Tangent space, Generalized flag variety, G-structure, Pullback (differential geometry), Affine differential geometry, Hilbert scheme, Spray (mathematics), Maurer-Cartan form, Affine focal set, Nonholonomic system, Poisson manifold, Associated bundle, Frame of a vector space, Evolute, Real projective space, K3 surface, Calabi conjecture, Winding number, Osculating circle, Tetrad formalism, Pushforward (differential), Dual curve, Double tangent bundle, Stiefel manifold, Ricci decomposition, Second fundamental form, Ruled surface, Monodromy, Calibrated geometry, Bogomolov-Miyaoka-Yau inequality, Metric signature, Multivector, Calculus of moving surfaces, Analytic torsion, Banach bundle, Tortuosity, Cartan's equivalence method, Reduction of the structure group, Projective connection, List of differential geometry topics, Invariant differential operator, Hitchin system, Courant bracket, Cartan formalism (physics), Courant algebroid. Excerpt: In mathematics, a manifold of dimension n is a topological space that near each point resembles n-dimensional Euclidean space. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot. The surface of the Earth requires (at least) two charts to include every point.Although near each point, a manifold resembles Euclidean space, globally a manifold might not. For example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of geographic maps: map projections of the region into the Euclidean plane. When a region appears in two neighbouring maps (in the context of manifolds they are called charts), the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical...
