DIFFERENTIAL GEOMETRY-II&TENSORS
Paper Code:
MAT224
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
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:
- Riemanian geometry & tensor calculus – Weatherburn – Cambridge University Press.
- Tensor Analysis, J.L.Bansal, JPH, Jaipur
- Tensor Analysis, Raj Bali, Navkar Publication, Ajmer.
Academic Year:
