Introduction to Smooth Manifolds

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

Type

Book

Intellectual base

Textbook

Year

2013

Pages cited

1-2

DOI

10.1007/978-1-4419-9982-5

ISBN

9781441999818

DOI registrant

Springer Nature

Link

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

Cited from

mathematics

Cited by

12 citations across 12 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.

    Supported in partVerified against the publisher's abstract

    Lee's abstract shows a graduate text whose scope includes embedded submanifolds, tangent vectors and tensors — the general framework — but it does not state that an embedded submanifold inherits gradients, integrals or a Euclidean metric.

    “Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology”

    Read by an automated reader; not a human review. How support was checked

• 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.
• Interior
  • Point-set topology — the home setting: the interior operator and its Kuratowski axioms, the exhaustive interior/boundary/exterior partition, and the closure relation A̅ = A° ∪ ∂A, the interior operator itself characterizing a topology. Real analysis — robust membership: interior points admitting a full-dimensional neighborhood, distinguished from boundary points where every neighborhood reaches the complement. Convex analysis — the relative interior: taken within a set's affine hull and guaranteed non-empty for a convex set, underwriting separation and duality theorems. Mathematical optimization — the toolkit selector: an interior optimum governed by unconstrained stationarity ∇f(x) = 0, a boundary optimum by Lagrange multipliers and KKT, and interior-point methods (logarithmic-barrier, primal-dual) navigating the relative interior to beat the simplex method's boundary traversal. Differential geometry — manifolds with boundary: the interior as the locus of points with Euclidean-neighborhood charts, distinct from the boundary stratum

    This sourceLee's Introduction to Smooth Manifolds, which defines the interior Int M of a manifold with boundary as the set of its interior points, distinct from the boundary ∂M.

    Supported in partVerified against the work's full text

    “similarly, its [interior], the set of all its [interior] points, is denoted by Int M.”

    Read by an automated reader; not a human review. How support was checked

• 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.

Verification

Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.

Does it back the claim? Read against the text for 2 of 12 citations: 2 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.

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