Introduction to Smooth Manifolds

Lee, J. M. (2013). Introduction to Smooth Manifolds. Springer.

Type

Book

Intellectual base

Textbook

Year

2013

DOI

10.1007/978-1-4419-9982-5

ISBN

9781441999818

Link

https://doi.org/10.1007/978-1-4419-9982-5

Cited from

mathematics

Cited by

11 citations across 11 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Coordinate-free
  • This blocks migration from an exact mathematical or empirical claim into a loose metaphor.
• Embedding
  • The payoff is inheritance of host machinery: \(\mathbb{R}^{2n}\) carries the entire apparatus of multivariable calculus — gradients, integrals, the Euclidean metric — none of which the abstract manifold had natively.

    This sourceStandard text establishing that a smoothly embedded submanifold inherits the ambient Euclidean apparatus — tangent vectors as derivatives, integration, the induced metric.

• Neighborhood
  • A topological manifold is a space in which every point has a neighborhood homeomorphic to ordinary flat Euclidean space.

    This sourceStandard reference: a manifold is a space in which every point has a neighborhood homeomorphic to flat Euclidean space, charts are local windows, transition maps are the glue condition, and the impossibility of a single global chart (e.g. for the sphere) is a gluing obstruction.

Domain-specific

• Collar neighbourhood
  • Collars give a controlled product geometry near boundaries, enabling gluing, doubling, extension and boundary-compatible constructions; existence depends on the relevant topological or smooth collaring theorem.
• Differential Structure
  • That maximal atlas is the smooth structure.
• Double (manifold)
  • Listed in the references but not attached to a specific claim.
• Integral curve
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Local diffeomorphism
  • A smooth map f:X→Y is a local diffeomorphism when every x in X has an open neighborhood U such that f(U) is open in Y and the restriction f|U:U→f(U) is a diffeomorphism.
• One-form
  • … the smooth manifold, cotangent bundle, regularity, pointwise linear functional, coordinate components and transformation law, pullback convention and any closed or exact claim are explicit. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Tangent bundle
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Transport of Structure
  • In differential topology, a homeomorphism \(h:X\to M\) can transport a smooth atlas from \(M\) to \(X\): charts \((U,\varphi)\) become \((h^{-1}U,\varphi\circ h)\), and the transition maps remain the original smooth transitions.

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