Overview

COURSE OUTLINE: The goal of this course is to introduce the student to the basics of smooth manifold theory. The course will start with a brief outline of the prerequisites from topology and multi-variable calculus. After that a large class of examples, including Lie groups, will be presented. The course will culminate with proof of Stokes' theorem on manifolds.

ABOUT INSTRUCTOR: Prof. Harish Seshadri is currently working as an Assistant Professor in the Department of Mathematics in IISC Banglore. He completed his M.Sc from IIT Kanpur and Ph.D from SUNY Stony Brook. He likes to work in Riemannian geometry (Einstein manifolds, Ricci flow, etc) and in questions related to invariant metrics in complex analysis.

Syllabus

Intro An introduction to smooth manifolds.
noc20 ma01 lec01 Basic linear algebra.
noc20 ma01 lec02 Multivariable calculus 1.
noc20 ma01 lec03 Multivariable calculus 2.
noc20 ma01 lec04 The derivative map.
noc20 ma01 lec05 Inverse Function Theorem.
noc20 ma01 lec06 Constant Rank Theorem.
noc20 ma01 lec07 Smooth functions with compact support.
noc20 ma01 lec08 Smooth manifold.
noc20 ma01 lec09 Examples of smooth manifolds.
noc20 ma01 lec10 Higher dimensional spheres as smooth manifolds.
noc20 ma01 lec11 Smooth maps.
noc20 ma01 lec12 Examples of smooth maps.
noc20 ma01 lec13 Tangent spaces I.
noc20 ma01 lec14 Tangent spaces 2.
noc20 ma01 lec15 Derivatives of smooth maps.
noc20 ma01 lec16 Chain rule on manifolds.
noc20 ma01 lec17 Dimension of tangent space 1.
noc20 ma01 lec18 Dimension of tangent space 2.
noc20 ma01 lec19 Derivative of inclusion map.
noc20 ma01 lec20 Basis of tangent space.
noc20 ma01 lec21 Inverse Function Theorem for manifolds.
noc20 ma01 lec22 Submanifolds.
noc20 ma01 lec23 Tangent space of a submanifold.
noc20 ma01 lec24 Regular Value Theorem.
noc20 ma01 lec25 Special linear group as a submanifold of the set of all square matrices.
noc20 ma01 lec26 Hypersurfaces.
noc20 ma01 lec27 Tangent spaces to level sets.
noc20 ma01 lec28 Vector fields 1.
noc20 ma01 lec29 Vector fields 2.
noc20 ma01 lec30 Vector fields 3.
noc20 ma01 lec31 Lie groups 1.
noc20 ma01 lec32 Lie groups 2.
noc20 ma01 lec33 Integral curve and flows 1.
noc20 ma01 lec34 Integral curve and flows 2.
noc20 ma01 lec35 Integral curve and flows 3.
noc20 ma01 lec36 Complete vector fields.
noc20 ma01 lec37 Vector fields and smooth maps.
noc20 ma01 lec38 Lie Brackets 1.
noc20 ma01 lec39 Lie brackets 2.
noc20 ma01 lec40 Lie brackets 3.
noc20 ma01 lec41 Lie algebras of matrix groups 1.
noc20 ma01 lec42 Lie algebras of matrix groups 2.
noc20 ma01 lec43 Exponential map.
noc20 ma01 lec44 Frobenius theorems.
noc20 ma01 lec45 Tensors and differential forms.
noc20 ma01 lec46 Tensors and differential forms 2.
noc20 ma01 lec47 Pull back form.
noc20 ma01 lec48 Symmetric Tensors.
noc20 ma01 lec49 Alternating Tensors 1.
noc20 ma01 lec50 Alternating Tensors 2.
noc20 ma01 lec51 Alternating Tensors 3.
noc20 ma01 lec52 Alternating Tensors 4.
noc20 ma01 lec53 Alternating Tensors 5.
noc20 ma01 lec54 Alternating Tensors 6.
noc20 ma01 lec55 Alternating Tensors 7.
noc20 ma01 lec56 Alternating Tensors 8.
noc20 ma01 lec57 Alternating Tensors 9.
noc20 ma01 lec58 Differential forms on manifolds 1.
noc20 ma01 lec59 Differential forms on manifolds 2.
noc20 ma01 lec60 The Exterior derivative 1.
noc20 ma01 lec61 The Exterior derivative 2.
noc20 ma01 lec62 The Exterior derivative 3.
noc20 ma01 lec63 The Exterior derivative 4.
noc20 ma01 lec64 The Exterior derivative 5.
noc20 ma01 lec65 Special classes of forms.
noc20 ma01 lec66 Orientation on manifolds 1.
noc20 ma01 lec67 Orientation on manifolds 2.
noc20 ma01 lec68 Orientation on manifolds 3.