The course is scheduled to meet from 10:30 to 11:45 am on Tuesdays and Thursdays in Lecture Hall 2 at CMI.
Some brief lecture notes are here.
Topics: concept of a manifold, cartography, coordinate charts, coordinate transformations or transition functions, atlas, topological and smooth manifolds, examples of manifolds; maps between manifolds: homeomorphisms and diffeomorphisms; submanifolds, immersions and embeddings, path connected and simply connected manifolds, smooth functions or scalar fields; vector fields: coordinate vector fields, tangent vectors, tangent space, transformation of components, Jacobian matrix, contravariant vector fields, integral curves of a vector field, Lie derivative of a scalar along a vector field, commutator of vector fields, Lie algebra of vector fields; covector fields or one-forms, Liouville 1-form, heat 1-form, differentials and coordinate one-forms, dual basis and pairing between vector fields and one-forms, cotangent space; tensors of rank 2, Poisson tensor, metric tensor, two-forms, wedge product and exterior derivative of one-forms, electromagnetic potential one-form and field strength two-form, symplectic two-form; mixed second-rank tensors; higher rank tensor fields and forms; pullback and pushforward, pullback or induced metric; exterior algebra, exterior derivative, Bianchi's identity, closed and exact forms, symplectic form and Hamiltonian vector field; orientable manifold, Riemannian volume form, integration of forms, manifold with boundary, Stokes' theorem; Geodesic equation: curve of extremal length; covariant derivative of a vector field on a Riemannian manifold, Christoffel connection coefficients, covariant derivatives of other tensors; curvature on a Riemannian manifold, Riemann-Christoffel curvature tensor, Ricci tensor and curvature scalar, sectional or Gaussian curvature, geodesic deviation or Jacobi equation and Riemannian curvature; Groups: cardinality, discrete and continuous groups, subgroup, homomorphism, kernel, isomorphism, automorphism; conjugation, conjugacy class, abelian and nonabelian groups; Lie groups, Matrix Lie groups; action of a group on a set, Lie group as a homogeneous manifold; Lie algebra of a Lie group; coset spaces, normal subgroup and quotient or factor group, commutator subgroup, simple and semisimple groups; direct product, semidirect product; permutation group, alternating group; circle group U(1); orthogonal group O(3); orthogonal Lie algebra, exponential map, Lie bracket and structure constants.
Problem set 1 is due by the beginning of the class on Tuesday Jan 9, 2024
Problem set 2 is due by the beginning of the class on Tuesday Jan 23, 2024
Problem set 3 is due by the beginning of the class on Tuesday Jan 30, 2024
Problem set 4 is due by the beginning of the class on Tuesday Feb 6, 2024
Problem set 5 is due by the beginning of the class on Tuesday Feb 13, 2024
Problem set 6 is due by the beginning of the class on Tuesday Feb 20, 2024
Problem set 7 is due by the beginning of the class on Tuesday Feb 27, 2024
Problem set 8 is due by the beginning of the class on Tuesday Mar 5, 2024
Problem set 9 is due by the beginning of the class on Tuesday Mar 12, 2024
Problem set 10 is due by the beginning of the class on Tuesday Mar 19, 2024
Problem set 11 is due by the beginning of the class on Tuesday Apr 2, 2024
Problem set 12 is due by the beginning of the class on Thursday Apr 18, 2024
Problem set 13 is due by the beginning of the class on Tuesday Apr 30, 2024
Problem set 14 is due by 1030am on Monday May 6, 2024