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The base field over which manifolds are defined is no longer assumed to be the real field: it can be any topological field (non-discrete to define differentiability for diff. manifolds). This allows to define easily complex manifolds, by setting the field to C .
If the scalar curvature is constant so that is cscK, then the associated holomorphy potential is a constant function, and the induced holomorphic vector field is the zero vector field. In particular on a Kähler manifold which admits no non-zero holomorphic vector fields, the only holomorphy potentials are constant functions and every extremal ...
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.
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
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, [1] is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
Differential geometry stubs (1 C, 115 P) Pages in category "Differential geometry" The following 200 pages are in this category, out of approximately 379 total.
Let M be a Banach manifold of class C r with r ≥ 2. As usual, TM denotes the tangent bundle of M with its natural projection π M : TM → M given by : (,). A vector field on M is a cross-section of the tangent bundle TM, i.e. an assignment to every point of the manifold M of a tangent vector to M at that point.
Given any curve c : (a, b) → S, one may consider the composition X ∘ c : (a, b) → ℝ 3. As a map between Euclidean spaces, it can be differentiated at any input value to get an element (X ∘ c)′(t) of ℝ 3. The orthogonal projection of this vector onto T c(t) S defines the covariant derivative ∇ c ′(t) X.
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