| Tensor algebra. |
Tensor product of vector spaces. (r,s)-Tensors, r-covariant and s-contravariant tensors. Tensor contraction. p-Alternating tensors. Wedge product. Orientation. Geometric interpretation of alternating tensors. |
| Differentiable manifolds. |
Locally Euclidean spaces. Topological manifolds. Locally finite covers, paracompact manifolds. Related C^k charts, equivalent atlas, C^k structures, differentiable manifold. Differentiable functions between manifolds, diffeomorphisms. Partition of unity. Tangent vector and tangent space. Differential of a function between manifolds. Vector fields and Lie derivative. (r,s)-Tensor fields over manifolds, r-covariant and s-contravariant. p-Differential forms. Exterior derivative of p-forms. Pullback of a differential p-form by a map between manifolds. |
| Riemannian geometry. |
Metric (2,0)-tensor field and (semi-) Riemannian manifolds. The special cases of the parameterized submanifolds of R^n with the standard metric, and the parameterized surfaces, with its induced metric, of R³ with the standard metric. Length of a curve on a manifold. Linear connections (Koszul connections) on manifolds, Christoffel symbols. First approach to the propagation of the linear connection to the layers of differential p-forms and (r,s)-tensor fields. Covariant derivation of fields along curves according to the linear connection. The Levi-Civita connection. Parallel translation along a curve. Geodesics. Torsion (2,0)-tensor field, and its expression on local charts. Riemann curvature (3,1)-tensor field, its expression in local charts, and its geometric interpretations. |
| Curvature. |
Bianchi's first identity. Riemann curvature (4,0)-tensor field over any Riemannian manifold, and properties. The special definition of curvature k_p, using the Riemann curvature (4,0)-tensor field, in the special case of a parameterized surface, with its induced metric, of R³ with the standard metric. Second fundamental form, normal curvatures, Weingarten map and Gaussian curvature K of surfaces, with its induced metric, of R³ with the standard metric. Coincidence, K=k_p, of the Gaussian curvature with the Riemannian curvature. Gauss's Theorema Egregium and Gauss equation for the curvature K. Definition of Gaussian curvature k_p for any 2-Riemannian manifold using the Riemann curvature (4,0)-tensor field. The geodesic exponential map. Definition of the Gaussian curvature k_p with the geodesic exponential map as sectional curvatures of any Riemannian manifold. Obtaining the sectional curvature with the Riemann curvature (4,0)-tensor field. Reconstruction of the Riemann curvature (4,0)-tensor field with the sectional curvatures. Ricci curvature (2,0)-tensor field as a contraction of the Riemann curvature (3,1)-tensor field, expression on local charts. Scalar curvature field as a contraction of the Ricci curvature (1,1)- tensor field, and expression on local charts. Geometric interpretation of the scalar curvature with the Riemann curvature (4,0)-tensor field, and its interpretation from the point of view of the difference with Euclidean geometry. Propagation of the linear connection to all tensor layers; covariant differential (r+1,s)-tensor field of another (r,s)-tensor field. Properties and expression in local charts, of the propagation to all tensor layers of the covariant derivation. Divergence (r,s-1)-tensor field from another (r,s)-tensor field. Divergence (r-1,0)-tensor field from another (r,0)-tensor field. Covariant differential (4,1)-tensor field of the Riemann curvature (3,1)-tensor field; and Bianchi's second identity. Divergence (1,0)-tensor field from the product of the scalar curvature field by the metric (2,0)-tensor field; exterior derivative of the scalar curvature field. Covariant differential (3,0)-tensor field of the metric (2,0)-tensor field. Divergence (1,0)-tensor field from the Ricci curvature (2,0)-tensor field; divergence (1,0)-tensor field from the Einstein (2,0)-tensor field. Theorem-definition of the Riemann curvature (3,1)-tensor field in relation to whether the kind of geometry on the manifold, locally verifies the postulates of Euclid's Geometry or not. |
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