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Flat and sharp are mutually inverse isomorphisms of smooth vector bundles, hence, for each p in M, there are mutually inverse vector space isomorphisms between T p M and T ∗ p M. The flat and sharp maps can be applied to vector fields and covector fields by applying them to each point. Hence, if X is a vector field and ω is a covector field,
In Hungarian Rhapsody No. 6, Franz Liszt takes the unusual step of changing the key from D-flat major to C-sharp major near the start of the piece, and then back again to B-flat minor. Maurice Ravel selected C-sharp major as the tonic key of "Ondine" from his piano suite Gaspard de la nuit.
Vector field; Tensor field; Differential form; Exterior derivative; Lie derivative; pullback (differential geometry) pushforward (differential) jet (mathematics) Contact (mathematics) jet bundle; Frobenius theorem (differential topology) Integral curve
Differential geometry finds applications throughout mathematics and the natural sciences. Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity, and subsequently by physicists in the development of quantum field theory and the standard model of particle physics.
In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces.
The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds: Let (M,g) be a closed smooth Riemannian manifold.
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by different amounts in different directions at that point.
Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces.