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Examples ofsmooth manifold
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We call a finite-dimensional compact (respectively non-compact)smoothmanifoldwithout boundary a closed (respectively open) manifold.
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Locally, asmoothmanifoldcan be expressed as a graph over its tangent plane.
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The foregoing set intersection is not asmoothmanifoldof displacements as will be explained in what follows.
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Throughout this paper, by a manifold (map, function) we mean asmoothmanifold(map, function).
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This is asmoothmanifoldwith boundary endowed with a smooth measure.
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Mathematically, spacetime symmetries are usually described by smooth vector fields on asmoothmanifold.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The circle is characterized as a manifold by being one-dimensional, compact and connected; here the characterization, as asmoothmanifold, is up to diffeomorphism.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Everysmoothmanifoldcan be embedded into a finite-dimensional linear space.
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This example is from Wikipedia and may be reused under a CC BY-SA license.A smooth path in asmoothmanifoldhas (at every point) the tangent vector, belonging to the tangent space (attached to this point).
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This example is from Wikipedia and may be reused under a CC BY-SA license.Important examples of vector bundles include the tangent bundle and cotangent bundle of asmoothmanifold.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The frame bundle of asmoothmanifoldis the one associated to its tangent bundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.But if the stabilizer groups change we can not expect asmoothmanifoldany longer.
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This example is from Wikipedia and may be reused under a CC BY-SA license.A more involved example comes from differential topology, in which we have the notion of asmoothmanifold.
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This example is from Wikipedia and may be reused under a CC BY-SA license.If the d-bar operator can be shown to be transverse to the zero-section, this moduli space will be asmoothmanifold.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Recall that a (smooth)manifoldhas a unique tangent bundle and a unique stable normal bundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Smooth (differentiable) functions, paths, maps, given in asmoothmanifoldby definition, lead to tangent spaces.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Asmoothmanifoldalways carries a natural vector bundle, the tangent bundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This leads, for example, to the notion of a smooth bundle map between smooth fiber bundles over asmoothmanifold.
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This example is from Wikipedia and may be reused under a CC BY-SA license.They are topological invariants associated to vector bundles on asmoothmanifold.
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This example is from Wikipedia and may be reused under a CC BY-SA license.These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.
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