Differential Geometry — Spring 2026

Course at the Kerala School of Mathematics. This course develops the differential geometry of smooth manifolds with an emphasis on Riemannian geometry and the geometry of surfaces. Topics include vector bundles, tensor fields, vector fields and Lie brackets, Riemannian metrics and connections, geodesics and curvature, the geometry of surfaces, and culminates in the Gauss–Bonnet theorem.

Instructor and contact

Instructor: Pranav Haridas
Office: F-5
Office hours: by appointment

Textbooks & recommended reading

  • Introduction to Smooth Manifolds, John M. Lee
  • Introduction to Riemannian Manifolds, John M. Lee
  • Differential Geometry of Curves and Surfaces, Manfredo do Carmo

Evaluation scheme

  • Assignments & quizzes: 30%
  • Midterm examination: 30%
  • Final examination: 40%

Lecture schedule

Date Lecture Hours Topics Resources
January 13 Lecture 1 2 Review of smooth manifolds, smooth maps, and tangent spaces, emphasizing geometric intuition and motivating the introduction of Riemannian structures.
Lecture 2 2 Vector bundles: definitions, local trivializations, transition functions, and fundamental examples including tangent and cotangent bundles.
Lecture 3 2 Tensor bundles constructed from vector bundles; tensor fields of type $(r,s)$ and their interpretation as multilinear geometric objects.
Lecture 4 2 Vector fields as derivations; integral curves and flows; examples illustrating geometric behavior of flows.
Lecture 5 2 Lie brackets of vector fields, algebraic properties, coordinate expressions, and interpretation via commutators of flows.
<Mid-semester Examination I> 2 Covers Lectures 1–5
Lecture 6 2 Distributions as subbundles of the tangent bundle; integrability; statement and examples related to the Frobenius theorem.
Lecture 7 2 Riemannian metrics: definitions, examples, induced notions of length, angles, orthogonality, and volume.
Lecture 8 2 Affine connections, torsion and metric compatibility, and the fundamental theorem guaranteeing the Levi–Civita connection.
Lecture 9 2 Covariant derivatives of vector and tensor fields; Christoffel symbols and computational techniques in local coordinates.
Lecture 10 2 Geodesics defined via connections; geodesic equations; geometric interpretation and examples.
<Mid-semester Examination II> 2 Covers Lectures 6–10
Lecture 11 2 The exponential map and normal coordinates; local structure of Riemannian manifolds and simplification of geometric quantities.
Lecture 12 2 Parallel transport along curves; path dependence and its relation to curvature and holonomy.
Lecture 13 2 The Riemann curvature tensor: definition, symmetries, identities, and sectional curvature.
Lecture 14 2 Curvature in low dimensions, with emphasis on Gaussian curvature and classical examples.
<Mid-semester Examination III> 2 Covers Lectures 11–14
Lecture 15 2 Surfaces in $\mathbb{R}^3$: immersions and embeddings, first fundamental form, induced metric, and area.
Lecture 16 2 Normal vector fields, second fundamental form, shape operator, and principal curvatures.
Lecture 17 2 Gaussian curvature as an intrinsic invariant; statement and consequences of Gauss’s Theorema Egregium.
Lecture 18 2 Curves on surfaces, geodesic curvature, and characterization of geodesics on surfaces.
<Mid-semester Examination IV> 2 Covers Lectures 15–18
Lecture 19 2 Global geometry of surfaces, total curvature, and motivation for the Gauss–Bonnet theorem.
Lecture 20 2 Connection 1-forms and curvature 2-forms; structure equations and differential-form viewpoint on curvature.
Lecture 21 2 Local Gauss–Bonnet theorem for geodesic polygons, angle defect, and explicit examples.
Lecture 22 2 Gauss–Bonnet theorem for compact surfaces without boundary and its relation to the Euler characteristic.
<Mid-semester Examination V> 2 Covers Lectures 19–22
Lecture 23 2 Extension of Gauss–Bonnet to surfaces with boundary, including boundary curvature terms and examples.
Lecture 24 2 Applications of Gauss–Bonnet to surface classification, geometry–topology interplay, and further directions.
Final Examination 3 Comprehensive final examination covering the entire course.

Course logistics & policies

  • Prerequisites: Smooth manifolds, tangent bundles, differential forms, and integration on manifolds.
  • Collaboration: Discussion of ideas is encouraged; submitted work must be individual.