Announcements

Lecture No	Date	Content	Corresponding Reading material
1	Sep. 28	A crude description of tensors and manifolds; some historical remarks about modern differential geometry; preliminary topics and background: Sets and their basic properties, union and intersection of collection of sets, Cartesian product of sets, relations, equivalence relations, parallelism for lines in a Euclidean plane	--
2	Sep. 30	Partitions of a set and their relation to equivalence relations; functions, image and inverse image of a subset under a function, domain and range of a function; composition of functions; everywhere-define, onto, and one-to-one functions, bijections; Equivalent sets and cardinal numbers	-
3	Oct. 05	Groups, permutation group of a set; vector spaces, subspace of a vector space; linear combination, Einstein summation convention, span, linear independence and basis	pages 1-4 of Wasserman's 2nd Ed.
4	Oct. 07	Basic properties of bases, expansion of vectors in a basis, dimension of a vector space; linear maps, kernel of a linear map, isomorphic vector spaces, classification of finite-dim. vector spaces, quotients of a vector space by its subspaces, fundamental theorem of homomorphisms for linear maps, projections	pages 6-8 of Wasserman's 2nd Ed.
5	Oct. 09	Matrix representation of linear maps, automorphisms as basis transformations, effect of basis transformations on matrix representation of linear operators.	pages 5 and 8-10 of Wasserman's 2nd Ed.
	Oct. 12	Quiz 1
6	Oct. 12	Vector space L(V,W) of linear maps between two vector spaces, algebraic dual of a vector space and dual basis, second dual of a vector space; vector space L(V1, V2,...,Vk,W) of multilinear maps	pages 11-19 of Wasserman's 2nd Ed.
7	Oct. 14	Inner products, nondegnerate bilinear maps, and their matric representations, Euclidean inner product, Minkowski's nondegenerate map, the Kronecker's delta function (natural pairing of V and V^*), tensor product of two vector spaces	pages 19-20 and 25-26 of Wasserman's 2nd Ed.
	Oct. 21	Quiz 2
8	Oct. 26	Tensor product of more than two vector spaces and their duals; tensors of type (r,s) for a finite-dimensional vector space; effect of a change of basis on the components of a type (r,s) tensor	pages 28-30 and 34-38 of Wasserman's 2nd Ed.
9	Nov. 02	Tensors with definite type as equivalence classes of the set of collection of their components. The tensor product of two tensors with definite type. The tensor algebra of a finite-dimensional vector space	-
10	Nov. 04	Symmetric and antisymmetric tensors, the symmetrization and antisymmetrization operations. The symmetric tensor product, the covariant (and contravariant) symmetric tensor algebras.	pages 87-92 of Bishop and Goldberg
11	Nov. 06	The antisymmetric tensor (wedge) product, the Grassmann algebra of a vector space and its properties.	pages 93-97 of Bishop and Goldberg
	Nov. 09	Midterm Exam 1
12	Nov. 11	Metric spaces; open, closed, bounded subsets of a metric spaces; interior, closure and boundary of a subset of a metric spaces; continuous functions between metric spaces, isometries, and metric space structures; topological spaces, metric topology, discrete topology; continuous functions between topological spaces, homeomorphisms, and topological structures	pages 102-103 of Wasserman's 2nd Ed. and pages 8-12 of Bishop and Goldberg
13	Nov. 16	Topological invariants; connected and compact (subsets of ) topological spaces, relative, product, and cofinite topologies; locally compact and Hausdorff topological spaces; basis of a topology, first and second countable topological spaces; refinements of a covering, locally finite coversings, and paracompact topological spaces	pages 103-105 of Wasserman's 2nd Ed. and pages 13-18 of Bishop and Goldberg
14	Nov. 18	Euclidean space, differentiability of a function mapping a Euclidean space to anther, partial derivatives and the Jacobi matrix, chain rule, Inverse function theorem, inverse function theorem; Locally Euclidean spaces, definition and some examples of a topological manifold	pages 15-45 of Spivak, and pages 47-48 of Tu
	Nov. 20	Quiz 3
15	Nov. 23	Atlases and their transition functions, An atlas for S^1, C^n-manifolds, C^n compatible charts and atlases, maximal C^n-atlases as C^n structures, C^n functions between two manifolds, C^n diffeomorphisms, different C^n manifolds which are diffeomorphic.	pages 48-51 of Tu & pages 108-110 + 116-119 of Wasserman
16	Nov. 25	Examples of manifolds: Open subsets of manifolds, GL(n,R), graphs of functions from open subsets of R^n to R^m; Product manifolds; Quotient topology, Real projective plane	pages 63-72 of Tu & pages 111-112 of Wasserman
17	Dec. 02	Directional derivative of a function f: R^n -> R; germs of smooth functions at a point in R^n, vectors in R^n as derivations of the algebra of germs of smooth functions at a point in R^n; Tangent vectors and the tangent space of a smooth manifold at a point; the coordinate representation of tangent vectors; the cotangent vectors and spaces; coordinate bases for the tangent and cotangent spaces, coordinate transformation rules for the components of tangent and cotangent vectors in a coordinate basis	pages 11-18 & 80-81 of Tu
	Dec. 04	Quiz 4
18	Dec. 07	Tensors and tensor fields on a manifold, differential forms and the exterior algebra, Maxwell's tensor and the Riemannian metric tensor; differential of a function mapping a manifold to another.	pages 78-80 of Tu and pages 116-121 of Bishop and Goldberg
19	Dec. 09	Coordinate representation of the differential of a function, Exterior Calculus: Exterior derivative of differential forms and its properties, interior product, exact and closed differential forms	165-172 of Bishop and Goldberg
	Dec.16	Midterm Exam 2
20	Dec. 23	Pullback of differential forms, integral curves of a smooth vector field, one-parameter group of diffeomorphisms and local flows generated by a smooth vector field	pages135-140 of Tu & pages 121-128 of Bishop & Goldberg
21	Dec. 25	Lie bracket, its coordinate representation and interpretation, Lie derivative of a smooth tensor field	pages141-146 of Tu & pages 128-138 of Bishop & Goldberg
22	Dec. 30	Parallelism and parallel transportation along a curve, Christoffel symbols and covariant derivative of a vector field, affine connection, covariant derivative of a tensor field, torsion tensor	pages 229-238 of Spivak Vol. 2
23	Jan. 04	Covariant t-derivative, geodesics on a manifold endowed with an affine connection, generating a tensor of type (r,s+1) by taking covariant derivative of a type (r,s) tensor along coordinate axes, the (Riemann) curvature tensor of an affine connection, the Ricci curvature tensor	pages 239-250 of Spivak Vol. 2
24	Jan. 06	Riemannian manifolds and the properties of the metric tensor, uniqueness of symmetric connection that is compatible with the metric, properties of the Riemann and Ricci curvature tensor of a Riemannian manifold. Einstein field equations.	pages 35-47 of DoCarmo
	Jan. 08	Quiz 5

Note: The pages from the textbook and the reading material listed above may not include some of the subjects covered in the lectures.