MA4205 : Differential Geometry (Spring 2025) | Academic Portal of IISER Kolkata

Details of MA4205 (Spring 2025)

Level: 4

Type: Theory

Credits: 4.0

Course Code	Course Name	Instructor(s)
MA4205	Differential Geometry	Sayan Bagchi

Syllabus
Basic Theory: Topological manifolds, examples, differentiable manifolds and maps, immersed and imbedded manifolds, submanifolds, partitions of unity, compact manifolds as closed submanifolds of Rⁿ. Tangent Space and Vector Fields: Definition of tangent vector as equivalence class of curves and derivations, tangent spaces and their mappings, tangent bundle, vector fields, integral curves, complete vector fields, Lie derivative and connection with Lie bracket of vector fields. Differential Forms and Integration: Wedge product, Exterior differentiation: definition, axiomatic treatment and coordinate invariance, closed and exact forms, review of classical line and surface integrals, orientation, Stokes theorem. de Rham Cohomology: Definition, elementary computation for compact orientable surfaces, computation of highest cohomology.

Syllabus

Basic Theory: Topological manifolds, examples, differentiable manifolds and maps, immersed and imbedded manifolds, submanifolds, partitions of unity, compact manifolds as closed submanifolds of Rⁿ.

Tangent Space and Vector Fields: Definition of tangent vector as equivalence class of curves and derivations, tangent spaces and their mappings, tangent bundle, vector fields, integral curves, complete vector fields, Lie derivative and connection with Lie bracket of vector fields.

Differential Forms and Integration: Wedge product, Exterior differentiation: definition, axiomatic treatment and coordinate invariance, closed and exact forms, review of classical line and surface integrals, orientation, Stokes theorem.

de Rham Cohomology: Definition, elementary computation for compact orientable surfaces, computation of highest cohomology.

Prerequisite
Analysis III (MA3101), Topology (MA3201)

References
Suggested Texts: Guillemin, V. and Pollack, A., Differential Topology, AMS Chelsea. Hirsch, M.W., Differential Topology, Springer. Kumaresan, S., A Course in Differential Geometry and Lie Groups, Hindustan Book Agency. Lee, J.M., Introduction to Smooth Manifolds, Springer-Verlag. Milnor, J,W., Topology from the Differentiable Viewpoint, Princeton University Press. Mukherjee, A., Topics in Differential Topology, Hindustan Book Agency. Spivak, M., A comprehensive Introduction to Differential Geometry, Vol. I, 3rd Edition, Publish or Perish. Tu, L.W., An Introduction to Manifolds, Universitext, Springer-Verlag.

References

Suggested Texts:

Guillemin, V. and Pollack, A., Differential Topology, AMS Chelsea.
Hirsch, M.W., Differential Topology, Springer.
Kumaresan, S., A Course in Differential Geometry and Lie Groups, Hindustan Book Agency.
Lee, J.M., Introduction to Smooth Manifolds, Springer-Verlag.
Milnor, J,W., Topology from the Differentiable Viewpoint, Princeton University Press.
Mukherjee, A., Topics in Differential Topology, Hindustan Book Agency.
Spivak, M., A comprehensive Introduction to Differential Geometry, Vol. I, 3rd Edition, Publish or Perish.
Tu, L.W., An Introduction to Manifolds, Universitext, Springer-Verlag.

Course Credit Options

Sl. No.	Programme	Semester No	Course Choice
1	IP	2	Not Allowed
2	IP	4	Not Allowed
3	IP	6	Not Allowed
4	MP	2	Not Allowed
5	MP	4	Core
6	MR	2	Not Allowed
7	MR	4	Not Allowed
8	MS	10	Elective
9	MS	4	Not Allowed
10	MS	6	Not Allowed
11	MS	8	Elective
12	RS	1	Elective
13	RS	2	Elective

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Details of MA4205 (Spring 2025)

Course Credit Options