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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,
See also multivariable calculus, list of multivariable calculus topics. Manifold. Differentiable manifold; Smooth manifold; Banach manifold; Fréchet manifold; Tensor analysis. Tangent vector
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.
A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at p only), and the geodesics through p are locally linear functions of t (the affine parameter).
In the language of differential geometry, this derivative is a one-form on the punctured plane. It is closed (its exterior derivative is zero) but not exact , meaning that it is not the derivative of a 0-form (that is, a function): the angle θ {\\displaystyle \\theta } is not a globally defined smooth function on the entire punctured plane.
In the study of mathematics, and especially of differential geometry, fundamental vector fields are instruments that describe the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions.
Here, () is the set of smooth vector fields on , and is the Lie derivative along the vector field . Examples of symplectomorphisms include the canonical transformations of classical mechanics and theoretical physics , the flow associated to any Hamiltonian function, the map on cotangent bundles induced by any diffeomorphism of manifolds, and ...