tangent space oor Katalaans
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The tangent space at any point on a symplectic manifold is a symplectic vector space.
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There are various equivalent ways of defining the tangent spaces of a manifold.
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The map assigning to x its tangent space defines a map from M to Gr(k, n).
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Each point of an n-dimensional differentiable manifold has a tangent space.
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This yields an equivalence between tangent spaces defined via derivations and tangent spaces defined via cotangent spaces.
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Riemannian manifolds are manifolds whose tangent spaces are endowed with a suitable inner product.
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The infinitesimal increments are then identified with vectors in the tangent space at a point.
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The tangent space is the generalization to higher-dimensional differentiable manifolds.
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This is a differential manifold with a Finsler metric, that is, a Banach norm defined on each tangent space.
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Any immersed oriented hypersurface in n-dimensional space has a contact lift to Z2n – 1 determined by its oriented tangent spaces.
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A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner.
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To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent spaces.
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Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as smooth (synonymous: non-singular), or else singular.
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If the local charts on a manifold are compatible in a certain sense, one can define directions, tangent spaces, and differentiable functions on that manifold.
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The tangent space of a Lie group can be given naturally the structure of a Lie algebra and can be used to classify compact Lie groups.
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Once the tangent spaces of a manifold have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving in space.
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At each point x in M, the tangent space to M can be considered as a subspace of the tangent space of Rn, which is just Rn.
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More generally, if a given manifold is thought of as an embedded submanifold of Euclidean space, then one can picture a tangent space in this literal fashion.
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The set of all velocities through a given point of space is known as the tangent space, and so df gives a linear function on the tangent space: a differential form.
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The Euclidean space itself carries a natural structure of Riemannian manifold (the tangent spaces are naturally identified with the Euclidean space itself and carry the standard scalar product of the space).
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All the tangent spaces of a manifold may be ‘glued together’ to form a new differentiable manifold with twice the dimension of the original manifold, called the tangent bundle of the manifold.
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(In order to do this, we have to translate the geometrical tangent space to M so that it passes through the origin rather than x, and hence defines a k-dimensional vector subspace.
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Intuitively speaking such a manifold M is a space that can be approximated near each point x by a vector space called its tangent space: the prototypical example is a smooth surface in R3.
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With this interpretation, the differential of f is known as the exterior derivative, and has broad application in differential geometry because the notion of velocities and the tangent space makes sense on any differentiable manifold.
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Like a Riemannian metric, a Hermitian metric consists of a smoothly varying, positive definite inner product on the tangent bundle, which is Hermitian with respect to the complex structure on the tangent space at each point.
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