Alphabeta MathHow statement and proof provenance work
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The induced tangent bundle chart
Definition
Let be a smooth chart on an -manifold . Every vector with has a unique expansion by Coordinate derivations form a basis of the tangent space. The induced tangent-bundle chart is
Depends on
Used by
- The pullback of the Euclidean tangent bundle is canonically trivial Corollary
- Formal immersion between smooth manifolds Definition
- Geodesic spray Definition
- Isolated zero and local index of a vector field Definition
- Nondegenerate zero of a vector field Definition
- A nowhere-zero vector field on an odd sphere Example
- The tangent bundle of the circle is a cylinder Example
- An embedding into Euclidean space gives a rank-(n-m) stable normal inverse Lemma
- An immersion into Rⁿ gives a rank-(n-m) representative of the stable normal bundle Lemma
- Opposite-index nondegenerate zeros cancel in a ball Lemma
- Stiefel-Whitney classes of the tangent bundle of real projective space Lemma
- Tangent-bundle chart transitions are smooth with smooth inverses Lemma
- The local index is additive under a transverse perturbation Lemma
- Assuming countable choice, an ambient metric identifies the two normal bundles Proposition
- Incompleteness is finite-time geodesic escape Proposition
- Smale-Hirsch makes rank reduction sufficient for Euclidean immersion in positive codimension Proposition
- The index of a zero is its zero-section intersection number Proposition
- Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure Theorem
- Metric completeness implies geodesic completeness Theorem
- The index of a nondegenerate vector-field zero Theorem
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
- Will J. Merry, Differential Geometry (standard reference, not scraped)
- Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)