This is a graduate-level introduction to differential manifolds (mainly on smooth manifolds).
Teaching assistants: 何玟辛，李进钊.
Final Score = 30% Homework + 30% Midterm Exam + 40% Final Exam.
September 23, 2024,
Topics: Introduction to this course, definition of manifold, Basic examples of manifolds.
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September 24, 2024,
Topics: Construct manifolds: product, open subset, quotient, basic examples of Lie groups.
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September 29, 2024,
Topics: Group action (by Lie groups), reduction (example: Grassmannian), definition of vector bundle (example: Möbius bundle).
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September 30, 2024,
Topics: Construct vector bundles (example: tangent bundle, cotangent bundle), sections (example: vector fields), Poincaré-Hopf theorem, directional derivative (of a function on a manifold), definition of connection.
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October 8, 2024,
Topics: Connection, Bracket (of vector fields), tensor (of vector spaces), universal property.
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October 14, 2024,
Topics: Tensor algebra, tensor bundle, tensor field (example: metric tensors), non-example of tensor fields (connection, bracket), Sym vs. Alt, wedge algebra (definition + basis).
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October 15, 2024,
Topics: Wedge product, Hodge star operation, wedge bundle, k-form (as sections of the wedge-k bundle), exterior derivative (formula), dd = 0, interior multiplication.
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October 21, 2024,
Topics: Examples in calculus of dd =0 (grad, curl, div), Maxwell's equations (via forms), definition of flowlines (examples: 2-torus sitting in \R^3, constructing symplectic matrices).
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October 22, 2024,
Topics: Completeness of vector fields, time-dependent vector fields, 1-parameter family of diffeomorphisms, equivalence to vector fields property, 2-parameter family of diffeomorphisms (example: related to bracket).
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October 28, 2024,
Topics: Pushforward and pullback (of diffeomorphisms), Lie derivative (with input vector fields and forms), computational examples of Lie derivative, Cartan's magic formula.
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October 29, 2024,
Topics: Definition of a submanifold (example: graph of a smooth map), submanifold is a smooth manifold itself, closed-subgroup Theorem, pushforward of a smooth map (between possibly different manifolds).
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November 4 and November 5, 2024 (lectured by Professor Yu LI),
Topics: Rank of a smooth map, critical point & value of a smooth map, regular point & value of a smooth map (example - higher genus closed surface), immersion and submersion (definition, example), Constant Rank Theorem (and many of its corollaries), embedding (definition), embedding of F = embedded submanifold, various embedding theorems.
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November 11, 2024,
Topics: Isometric embeddings, covariant/contravariant functors (example: pushforward and pullback), (vector) bundle map, short exact sequence of vector bundles (example: normal bundle and conormal bundle), transversality of submanifolds, transversality of maps (example: fiber bundle).
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November 12, 2024,
Topics: Fiber bundle (universal property), "bump" function (construction), partition of unity (P.O.U.), three applications: bump function over any closed subset, smooth extension, "compact" version of Whitney embedding theorem.
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November 18, 2024,
Topics: Definition of integration, (no boundary version of) Stokes' theorem, manifold with boundary, orientability of the boundary, examples of manifold with boundary (sublevel set), Brouwer's fixed point theorem (application: eigenvalue of positive-entry matrix).
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November 19, 2024,
Topics: Orientation, induced orientation on the boundary, computation of integration via parametrization, Stokes' Theorem (example: recover the Green formula and the Gauss formula in calculus), definition of divergence (on a manifold), Divergence theorem (as a generalization of the Gauss Theorem).
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November 25, 2024,
Open-book exam, covering all materials in notes above until November 19 + HW1, HW2, HW3, HW4.
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November 26, 2024,
Topics: Meaning of divergence, applications of Stokes' Theorem: (1) orientation characterized by H^n(M; \R); (2) Moser's trick (constructing diffeomorphism); (3) homotopy invariant of closed 1-form (which leads to the observation that forms can detect topology), Recollection of elements in Lie group so far.
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December 2, 2024,
Topics: Left invariant vector field, definition of Lie algebra (example: gl(n,\R)), abelian Lie algebra structure, Lie group representation (example: adjoint representation Ad), Lie algebra representation (example: pushforward of Ad = ad), Ado-Iwasawa Theorem.
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December 3, 2024,
Topics: Exponential map, local diffeomorphic property of exp, three relations between Lie group and Lie algebra: (1) pushforward of a Lie group homomorphism determines the Lie group homomorphism; (2) Lie's third theorem (classifying simply connected Lie group); (3) connected Lie group is abelian if and only if its Lie algebra is abelian. Compute exp on GL(n,\R), and Baker–Campbell–Hausdorff formula.
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December 9, 2024,
Topics: Structure constants, Maurer-Cartan 1-forms and equation, connection 1-forms and connection matrix, curvature 2-forms and curvature matrix, curvature (0,4)-tensor, sectional curvature, bi-invariant metric on a compact Lie group (construction via averaging trick), represent action by connection via Lie bracket.
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December 10, 2024,
Topics: (On a Lie group with a bi-invariant metric) represent curvature via Lie bracket, a deep result on sectional curvature of a Lie group equipped with a bi-invariant metric, basic concepts in homological algebra: cochain complex, cohomology groups, chain map, short exact sequence of cochain complex (and its induced long exact sequence of cohomology groups), connecting morphism, two models of "relative" de Rham cochain complex, (chain) homotopic between two chain maps.
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December 16 and December 17, 2024, NO CLASS (LECTURER OUT FOR A CONFERENCE).
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December 23, 2024,
Topics: Topological homotopy induces chain homotopy (homotopy invariance of de Rham cohomology groups), Eilenberg-Steenrod axioms (definition of a cohomology theory), Mayer-Vietoris sequence (from Elienberg-Steenrod axioms), computational examples from MV-sequence (S^1, S^n, CP^n), finite good open cover (implies finite-dimension property of de Rham cohomology groups), K\"unneth formula (example: T^n), connected sum formula (example: \Sigma_{g \geq 2}), cup product structure (of de Rham cohomology classes).
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December 24, 2024,
Topics: compactly supported de Rham cohomology (definition, computational example \R^1), failure of homotopy invariance (to fix it: assuming map to be proper; in special cases consider pushforward of forms), Poincar\'e duality (for orientable manifold), degree of a smooth map (example: antipotal map of S^n), general hairy ball theorem, extension implies degree-zero.
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December 30, 2024,
Topics: Hodge *-operations, Hodge-Laplace operator, \delta-operator, Hodge theorem, Proof of Poincar\'e duality (via Hodge theorem), Distribution (and examples), contact structure (i.e., completely non-integrable hyperplane field).
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December 31, 2024,
Topics: Frobenious integrability theorem (both involutive version via vector fields and differential ideal version via forms), Morse-Sard's Theorem, applications (including proof of the degree of a smooth map is always an integer); [not reached but included in the notes: non-degeneracy of a critical points, Morse function, generic existence of Morse functions].
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January 6, 2025,
Closed-book exam, covering all material lectured in this semester (including HWs), focusing more on the second half of the semester (starting from November 26, 2024).
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