DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-09-01

How 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.

Secant and tangent direction maps of a Euclidean embedding

Definition

Let $N \geq 1$ and let $f : M \to R^{N}$ be a smooth embedding.

The secant direction map of $f$ is $σ_{f} : (M \times M) ∖ Δ_{M} \to S^{N - 1}, σ_{f} (p, q) : = \frac{f (q) - f (p)}{∥ f (q) - f (p) ∥},$ where $Δ_{M}$ is the diagonal and the norm comes from the Euclidean inner product on $R^{N}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

The tangent direction map of $f$ is $τ_{f} : T M ∖ 0_{M} \to S^{N - 1}, τ_{f} (p, v) : = \frac{d f_{p} (v)}{∥ d f_{p} (v) ∥} .$ This is well defined because $f$ is an immersion, so $d f_{p} (v) \neq 0$ for every nonzero tangent vector $v$ (Smooth embeddings). The construction uses only the punctured tangent fibres; whenever its smooth-manifold structure is needed, it is obtained locally by deleting the zero section in tangent-bundle charts.

Depends on

Dependency tree · 2 levels

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 6