Geometry 4

0) COMPLEMENTS OF TOPOLOGY
- Metric Spaces
Metric spaces and distance; examples (bounded metrics, the uniform convergence metric on C^0([a, b]), the L^1 distance on su C0([0, 1])). Continuity in metric spaces. Sequences and limits, subsequences, limit points. Review of the first countability axiom and properties of spaces satisfying it. Sequences in metric spaces, Cauchy sequences, completeness; subspace-induced metric; isometries.

- Compactness, Countability Axioms, Paracompactness, Separability
Equivalent notions of compactness in metric spaces: Fréchet compactness, sequential compactness. Totally bounded metric spaces, Lebesgue number of an open cover; characterization of compactness in a metric space (X,d) and its relationship with completeness. Lindelöf compactness; paracompactness. Separability and countability axioms.

DIFFERENTIABLE MANIFOLDS
1) Differentiable Manifolds
Topological manifolds and examples (graphs of continuous functions, spheres, product manifolds); topological properties of topological manifolds; differentiable structures and atlases; differentiable manifolds and examples (Euclidean spaces, finite-dimensional vector spaces, spaces of matrices, spheres, level sets of functions from
^m to R, product manifolds). Topological manifolds with boundary, differentiable manifolds with boundary. Construction of the real projective space ^m_ .

2) Maps Between Differentiable Manifolds
Smooth functions, coordinate representations; smooth maps; continuity of smooth maps and equivalent characterizations; examples of smooth maps; diffeomorphisms; partitions of unity, existence of bump functions.

3) The Tangent Space
Algebra of germs of smooth functions at a point; tangent vectors and their relation to directional derivatives in Euclidean space; _ as an -dimensional vector space; the differential (pushforward) and its properties; canonical basis of the tangent space at a point; tangent space to an open set of a manifold; smooth curves on a manifold, tangent vector to a curve and coordinate representation; tangent space of a finite-dimensional vector space; cotangent space and dual basis; pullback (or codifferential); local expression of the pushforward. Coordinate changes.

4) Rank Theorem, Inverse Function Theorem and Applications; Immersions, Submersions, Embeddings and Submanifolds
Inverse function theorem; rank of a map; rank theorem; immersions and submersions; theorems in the context of manifolds; topological and smooth embeddings; sufficient conditions for a smooth injective immersion to be an embedding; every smooth immersion is locally an embedding. Smooth submanifolds: embedded submanifolds, relation between smooth embeddings and embedded submanifolds, graphs of smooth maps as embedded submanifolds; local k-slice condition, level sets of constant-rank smooth maps; regular/critical points and values of smooth maps, the fiber theorem; local defining maps. Immersed submanifolds, local and global parametrizations. Restriction of maps to submanifolds; tangent space to a submanifold: characterization in the embedded case and via local defining maps.

5) Tangent and Cotangent Bundles - Vector Fields
The tangent bundle: topology, differentiable structure, smooth atlas induced by the base manifold; pushforward as a global map. Vector fields as sections of the tangent bundle, components in a local chart, characterization via their action on smooth functions; vector fields along subsets of a manifold; algebraic structure of the module
X(M) over the ring of smooth functions; Lie bracket and its properties, Lie algebras; brief mention of the Lie algebra of a Lie group, ψ-related vector fields, left-invariant vector fields; integral curves, local flow of a vector field; geometric meaning of the Lie bracket. Cotangent bundle and 1-forms, pullback of 1-forms and its properties, differential of a smooth function.

6) Vector Bundles (Overview)
Vector bundles, smooth vector bundles, local trivializations; examples: product bundle, Möbius bundle, tangent bundle; composition of smooth local trivializations and transition functions; local/global sections and smooth sections, restriction of a vector bundle, examples of sections; local and global frames, frames associated to local trivializations; parallelizable manifolds.

7) Multilinear Algebra and Tensor Fields
Multilinear maps and examples; tensor product of multilinear maps; basis for the space of multilinear maps; decomposable tensors; covariant, contravariant, and mixed-type tensors on a vector space V, bases induced by a choice of basis for V; symmetric tensors: symmetrization, symmetric product and properties; alternating (skew-symmetric) tensors on V. Tensors, tensor fields, and tensor bundles on manifolds; characterization of smooth covariant -tensors via multilinearity on smooth functions; symmetric tensor fields; pullback of tensor fields under smooth maps and its properties.

8) Differential Forms and the Exterior Derivative
Algebra of alternating tensors, antisymmetrization; elementary alternating tensors and bases for \Lambda^k(V^*);n-forms and endomorphisms; wedge product and its properties; decomposable covectors; exterior algebra. Differential forms on manifolds, pullback by smooth maps, wedge product and coordinate changes; line integrals of 1-forms and properties, fundamental theorem of line integrals; exact, closed, conservative 1-forms; exterior derivative operator: Euclidean case and properties, manifold case; exterior derivative and pullback; closed and exact k-forms; invariant formulation of the exterior derivative.

9) Oriented Manifolds and Integration on Manifolds
Equioriented bases of vector spaces, orientation, relation between orientation and alternating tensors; orientability and orientations on manifolds, oriented manifolds, orientation forms; orientation of hypersurfaces, orientation of the boundary of a manifold with boundary. Integration of differential forms: Euclidean case, integration on manifolds, independence of cover and partition of unity; properties of integrals of forms; Stokes' theorem.

10) Overview of Riemannian Manifolds
Riemannian metrics, metric induced by an immersion; examples (flat metric, polar coordinates, helicoid metric); length of a vector, angle between vectors; orthonormal frames; length of a piecewise smooth curve, invariance under reparametrization; distance on a Riemannian manifold; Riemannian manifolds as metric spaces and topology induced by the distance; isomorphism between tangent and cotangent spaces, musical isomorphisms, relation between differential and gradient.