Differential Geometry-I

Paper Code: 
MAT124
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to –

  1. Acquaint with the fundamentals of differential geometry primarily by focusing on the theory of curves and surfaces in three spaces. 
  2. Compute quantities of geometric interest such as curvature, as well as develop a facility to compute in various specialized systems, such as semi geodesic coordinates or ones representing asymptotic lines or principal curvatures. 
  3. Learn about tangent spaces, Surfaces, Gauss map, Geodesics on surfaces and curvature of plane curve.

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

MAT124

Differential Geometry-I (Theory)

The students will be able to –

 

CO19: Compute quantities of geometric interest such as curvature, as well as develop a facility to compute in various specialized systems, such as semigeodesic coordinates or ones representing asymptotic lines or principal curvatures.

CO20: Develop arguments in the geometric description of curves and surfaces in order to establish basic properties of geodesics, parallel transport, evolutes, minimal surfaces  

CO21: Determine and calculate curvature of curves in different coordinate systems.

CO22: Analyse the concepts and language of differential geometry and its role in modern mathematics

CO23: Explain the normal curvature of a surface, its connection with the first and second fundamental form and Euler’s theorem

CO24: Explain the concepts surfaces of revolution with constant negative and positive Gaussian curvature.

 

Approach in teaching:

Interactive Lectures, Discussion, Tutorials, Reading assignments, Demonstration, Team teaching, PowerPoint presentations.

 

Learning activities for the students:

Self learning assignments, Effective questions, Seminar presentation, Posters and Charts preparation.

Class test, Semester end examinations, Quiz, Assignments, Presentation, Individual

 

Unit I: 
I
15.00
Theory of curves: Space curves, Tangent, Contact of curve and surface, Osculating plane, Principal normal and binormal, Curvature, Torsion, Serret-Frenet's formulae.
 
Unit II: 
II
15.00
Osculating circle and osculating sphere, Existence and uniqueness theorems for space curves, Bertrand curves, Involutes, Evolutes. 
 
Unit III: 
III
15.00
Envelops and edge of regression, Ruled surfaces, Developable surfaces, Tangent plane to a ruled surface, Necessary and sufficient condition that a surface ζ=f (ξ,η) should represent a developable surface, Metric of a surface: First, second and third fundamental forms.
 
Unit IV: 
IV
15.00
Fundamental magnitudes of some important surfaces, Orthogonal trajectories, normal curvature, Meunier's theorem, Principal directions and principal curvatures, First curvature, Mean curvature, Gaussion curvature, Umbilics.
 

 

Unit V: 
V
15.00
The radius of curvature of any normal section at an umbilic on z = f (x,y), Radius of curvature of a given section through any point on z = f(x,y), Lines of curvature, Principal radii, Relation between fundamental forms, Curvature of the normal section.
 
Essential Readings: 
  • J.L. Bansal and P.R. Sharma, Differential Geometry, Jaipur Publishing House Jaipur, 2013.
  • P.P. Gupta and G.S. Malik, Differential Geometry, Pragati Prakashan, Meerut, 2012.
  • Prasun Kumar Nayak, Tensor Calculus and Differential Geometry, PHI Learning Pvt. Ltd., 2012.
  • Raj Bali, Differential Geometry, Navkar Publication, Ajmer, 2012.
References: 
  • T. J. Willmore,   An Introduction to Differential Geometry, Oxford University Press, London, 1997.
  • Erwin Kreyszig, Differential Geometry, Dover Publishing, 2003.
  • H.K. Pathak and J.P. Chauhan, Differential Geometry, Shiksha Sahitya Prakashan, 2012.
  • Clifford Henry Taube’s, Differential Geometry, Oxford university press, 2011.
  • B. D. Neill, Elementary Differential Geometry, Academic Press, London, 2006.
  • Millan and G. D. Parker, Elements of Differential Geometry, PHI, 1977.
  • D. Somasundaram, Differential Geometry, Narosa Publishing House, New Delhi, 2005.
Academic Year: 
2022-2023
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