Differential Geometry-II&Tensors
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.
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.
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.
Christoffel symbols and their properties, Covariant differentiation of tensors. 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. Einstein tensor, Flat space, Isotropic point, Schur's theorem.
1. Weatherburn – Riemanian geometry & tensor calculus –Cambridge University Press.
2. J.L.Bansal, Tensor Analysis, JPH, Jaipur
3. Raj Bali, Tensor Analysis, Navkar Publication, Ajmer.
1. Clifford Henry Taube’s, Differential geometry ,Oxford university press (2011).
2. Dirk J. Struik, Lectures on Classical Differential Geometry, Second Edition,Addison Wesley Publishing Company, London, (1961).
3. Quddus Khan, Differential Geometry of Manifolds, PHI Learning, 2012.
