Page 4 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

SYLLABUS

Complex Analysis and Differential Geometry

Objectives:
To emphasize the role of the theory of functions of a complex variable, their geometric properties and indicating some applications.

Introduction covers complex numbers; complex functions; sequences and continuity; and differentiation of complex functions.
Representation formulas cover integration of complex functions; Cauchy's theorem and Cauchy's integral formula; Taylor series; and
Laurent series. Calculus of residues covers residue calculus; winding number and the location of zeros of complex functions; analytic
continuation.
To understand classical concepts in the local theory of curves and surfaces including normal, principal, mean, and Gaussian

curvature, parallel transports and geodesics, Gauss's theorem Egregium and Gauss-Bonnet theorem and Joachimsthal's theorem,
Tissot's theorem.

Sr. No. Content

1 Sequences and functions of complex variables, Continuity, Differentiability
2 Analytic functions, Cauchy-Riemann equations, Cauchy’s theorem and Cauchy’s

integral formula, Conformal mappings, Bilinear transformations
3 Power Series, Taylor’s series and Laurent’s series, Singularities, Liouville’s

theorem, Fundamental theorem of algebra.

4 Cauchy’s theorem on residues with applications to definite integral evaluation,
Rouche’s theorem, Maximum Modulus principle and Schwarz Lemma

5 Notation and summation convention, transformation law for vectors, Knonecker
delta, Cartesian tensors, Addition, multiplication, contraction and quotient law of

tensors,

6 Differentiation of Cartesians tensors, metric tensor, contra-variant, Covariant and
mixed tensors, Christoffel symbols, Transformation of christoffel symbols and

covariant differentiation of a tensor,

7 Theory of space curves: - Tangent, principal normal, binormal, curvature and
torsion, Serret-Frenet formulae Contact between curves and surfaces, Locus of

Centre of curvature, spherical curvature
8 Helices, Spherical indicatrix, Bertrand curves, surfaces, envelopes, edge of

regression, Developable surfaces, Two fundamental forms

9 Curves on a surface, Conjugate direction, Principal directions, Lines of Curvature,
Principal Curvatures, Asymptotic Lines, Theorem of Beltrami and Enneper,

Mainardi-Codazzi equations,
10 Geodesics, Differential Equation of Geodesic. Torsion of Geodesic, Geodesic

Curvature, Geodesic Mapping, Clairaut.s theorem, Gauss- Bonnet theorem,

Joachimsthal.s theorem, Tissot.s theorem