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DIFFERENTIAL GEOMETRY-II&TENSORS [1]

Paper Code: 
MAT 224
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Unit I: 
I
15.00

Gauss's formulae, Gauss's characteristic equation, Weingarten equations, Mainardi-Codazzi equations. Fundamental existence theorem for surfaces, Parallel surfaces, Gaussian and mean curvature for a parallel surface, Bonnet's theorem on parallel surfaces.

Unit II: 
II
15.00
Geodesics, Differential equation of a geodesic, Single differential equation of a geodesic, Geodesic on a surface of revolution, Geodesic Curvature and Torsion, Gauss-Bonnet Theorem.
 
Unit III: 
III
15.00

Tensor Analysis-Contravariant and Covariant tensors, Kronecker delta, Symmetric tensors, Skew Symmetric tensors, Quotient law of tensors, Relative tensor. Riemannian space. Metric tensor, Indicator, Permutation symbols and Permutation tensors.

 
Unit IV: 
IV
15.00
Christoffel symbols and their properties, Covariant differentiation of tensors. Ricci's theorem, Intrinsic derivative, Differential equation of geodesic of a metric, Geodesic coordinates.
 
Unit V: 
V
15.00
Reimann-Christoffel tensor and its properties. Covariant curvature tensor, Einstein space. Bianchi's identity. Einstein tensor, Flat space, Isotropic point, Schur's theorem.
 
Essential Readings: 
  1. Riemanian geometry & tensor calculus – Weatherburn – Cambridge University Press.
  2. Tensor Analysis, J.L.Bansal, JPH, Jaipur
  3. Tensor Analysis, Raj Bali, Navkar Publication, Ajmer.
Academic Year: 
2014-2015 [2]

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Links:
[1] https://maths.iisuniv.ac.in/courses/subjects/differential-geometry-iitensors-1
[2] https://maths.iisuniv.ac.in/academic-year/2014-2015