Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The tangent space of an n-manifold has dimension n
Statement
If is a smooth -manifold and , then is an -dimensional real vector space.
Facts & Assumptions
Given: A smooth -manifold and a point .
In any smooth chart around , the coordinate derivations form a basis of (Coordinate derivations form a basis of the tangent space).
Proof
Choose a smooth chart around ; its coordinate basis has exactly vectors.
By [L1], those coordinate derivations form a basis of .
Therefore .
Depends on
Used by
- The space of Jacobi fields along a geodesic has dimension two n Corollary
- Conjugate points along a geodesic and their multiplicity Definition
- Conjugate antipodes on the round sphere Example
- The pointwise norm on a tangent space is smooth off the zero vector Lemma
- Unit spheres in finite-dimensional normed spaces are sequentially compact Lemma
- Conjugate instants are isolated unless the geodesic is constant Proposition
- Conjugate points and multiplicity are invariant under affine reparametrization Proposition
- Integral manifolds have the distribution dimension Proposition
- A geodesic does not minimize past its first conjugate point Theorem
- Conjugate points are critical values of the exponential map along the geodesic Theorem
- Existence and uniqueness of jacobi fields from initial data Theorem
- Index lemma Theorem
- The tangent space at the identity is a Lie algebra Theorem
Dependency tree · two levels
5 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)