Differential Geometry-II & Tensor Analysis

Paper Code: 
24MAT224
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to –

  1. Understand the role of tensors in differential geometry.
  2. Understand the interpretation of the curvature tensor, Geodesic curvature, Gauss and Weingarten formulae.
  3. Learn and apply problem-solving with differential geometry to diverse situations.

 

Course Outcomes: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

24MAT

224

Differential Geometry-II & Tensor Analysis (Theory)

CO54: Describe the asymptotic line and Gauss formula, explored Weingarten formulae and existence theorem.

CO55: Explain the concept of parallel surfaces and their properties. Investigate geodesics on surfaces of revolution and understand their geometric characteristics.

CO56: Express geodesic curvature and torsion regarding Gauss coefficients and analyse its geometric interpretation. Explore the Gauss-Bonnet Theorem (Joachimsthal theorem).

CO57: Apply the fundamental concepts of tensors, including their geometric interpretation and transformation properties.

CO58: Explore advanced topics in tensor analysis, such as Riemannian geometry, differential geometry and applications in engineering and physics.

CO59: Contribute effectively in course-specific interaction.

Approach in teaching:

Interactive Lectures, Discussion, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions,  Topic  presentation, Assigned tasks

 

 

Quiz, Class Test, Individual projects,

Open Book Test, Continuous Assessment, Semester End Examination

 

 

 

Unit I: 
Asymptotic lines and Gauss's Equations :
15.00

Definition, Differential equation of a asymptotic lines, Theorems on asymptotic lines, Curvature and torsion of an asymptotic line, Gauss's formulae, Gauss's characteristic equation, Weingarten equations, Mainardi-Codazzi equations.

Unit II: 
Parallel surfaces and Geodesics:
15.00

Fundamental existence theorem for surfaces, Parallel surfaces, Gaussian and mean curvature for a parallel surface, Bonnet's theorem on parallel surfaces, Geodesics: Definition, General differential equation of a geodesic on a surface , Single differential equation of a geodesic, Geodesic on a surface of revolution.

 

Unit III: 
Geodesic curvature :
15.00

 Geodesic curvature, Geodesic curvature in form of Gauss coefficient, Bonnet’s formula for Geodesic curvature and torsion of a Geodesic, Normal angle, Geodesic torsion, Gauss-Bonnet Theorem (Joachimsthal theorem).

 

Unit IV: 
Tensor Analysis:
15.00

Definition, Kronecker delta, Symmetric tensor, Skew Symmetric tensor, Quotient law of tensor, Relative tensor, Metric tensor, Indicator, Permutation symbols and Permutation tensor, Christoffel symbols and their properties, Covariant differentiation of tensor.

 

Unit V: 
Geodesic and Reimann-Christoffel tensor :
15.00

Ricci's theorem, Intrinsic derivative, Differential equation of geodesic of a metric, Geodesic coordinates, Reimann-Christoffel tensor and its properties, Covariant curvature tensor, Einstein space, Bianchi's identity.

 

Essential Readings: 
  • J.L. Bansal, Differential Geometry, Jaipur Publishing House, Jaipur, 2014.
  • J.L. Bansal, Tensor Analysis, Jaipur Publishing House, Jaipur, 2012.
  • P.P. Gupta and G.S. Malik, Differential Geometry, Pragati Prakashan, 2012.
  • Raj Bali, Tensor Analysis, Navkar Publication, Ajmer, 2012.

SUGGESTED READING

  • B.D. Neill, Elementary Differential Geometry, Academic Press, London, 2006.
  • Clifford Henry Taube’s, Differential Geometry, Oxford University Press, 2011.
  • Dirk J. Struik, Lectures on Classical Differential Geometry, Addison Wesley Publishing Company, London, 2003.
  • D. Somasundaram, Differential Geometry, Narosa Publishing House, New Delhi, 2005.
  • Erwin Kreyszig, Differential Geometry, Dover Publishing, 2003.
  • H.K. Pathak and J.P. Chauhan, Differential Geometry, Shiksha sahitya Prakashan, 2012.
  • Millan and G.D. Parker, Elements of Differential Geometry, PHI, 1977.
  • Nirmala Prakash, Differential Geometry, Tata McGraw Hill, 1981.
  • Prasun K. Nayak, Tensor Calculus and Differential Geometry, PHI Learning Private Limited, 2012.

e- RESOURCES

 

JOURNALS

 

Academic Year: 
2024-2025
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