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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,
The metric tensor (,) induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat ♭ and sharp ♯. A section A ∈ Γ ( T M ) {\displaystyle A\in \Gamma (TM)} corresponds to the unique one-form A ♭ ∈ Ω 1 ( M ) {\displaystyle A^{\flat }\in \Omega ^{1}(M)} such that for all sections X ∈ Γ ( T M ...
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.
Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper. [1]
where , are orthonormal vector fields on , ~, ~ their horizontal lifts to , [,] is the Lie bracket of vector fields and is the projection of the vector field to the vertical distribution. In particular the lower bound for the sectional curvature of N {\displaystyle N} is at least as big as the lower bound for the sectional curvature of M ...
Even so, Johannes Brahms still felt the need to rewrite his C-sharp minor piano quartet in C minor, which was published as Piano Quartet No. 3 in C minor, Op. 60. [citation needed] The last intermezzo from his Three Intermezzi for piano, Op. 117 is in C-sharp minor. Alkan composed the second movement (Adagio) for Concerto for Solo Piano in C ...
In mathematics, an exotic is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space. The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.