## Manifolds and Differential GeometryDifferential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature. At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle. This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory. The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hyper-surfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations. The first chapters of the book are suitable for a one-semester course on manifolds. There is more than enough material for a year-long course on manifolds and geometry. |

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action atlas basis called cohomology commutes compact components connected consider coordinate Corollary countable covariant derivative curvature deﬁned Deﬁnition denote diffeomorphism differential domain element equation example Exercise expp ﬁber ﬁrst ﬁxed formula frame ﬁeld geodesic geometry given GL(V global homomorphism integral curve integral manifold isometry isomorphism Lemma Let f Lie algebra Lie group Lie subgroup linear map map f matrix metric module morphism n-manifold normal notation notion obtain open set open subset oriented orthonormal principal bundle Proof Proposition pull-back Qk(M quotient Recall regular submanifold restriction Riemannian manifold scalar product semi-Riemannian manifold Show smooth curve smooth function smooth manifold smooth map smooth structure submanifold subspace Suppose tangent bundle tangent map tangent space tangent vector tensor fields Theorem timelike topology trivial unique vector bundle vector field vector space zero