Kobayashi-Nomizu — Foundations of Differential Geometry, Vol. I (Fast Track 3.18) — Audit + Gap Plan

Book: Shoshichi Kobayashi, Katsumi Nomizu, Foundations of Differential Geometry, Volume I. Interscience Tracts in Pure and Applied Mathematics 15, Interscience Publishers (Wiley), 1963. xi + 329 pp. Reprinted Wiley Classics Library 1996 (ISBN 0-471-15733-3). The canonical English-language reference for connection-theoretic differential geometry.

Fast Track entry: 3.18 (KN Vol. I is the cited anchor for connection-and-curvature units across the Codex modern-geometry chapter; KN Vol. II is its own Fast Track entry 3.19, deferred to a separate audit).

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units so that Foundations of Differential Geometry Vol. I (KN-I hereafter) is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

Audit mode: REDUCED. No local PDF is available (reference/textbooks-extra/ and reference/fasttrack-texts/03-modern-geometry/ checked — only the cover-image stub reference/fast-track/images/Kobayashi-Nomizu-1-…jpg is present; KN-I is copyright-protected and was not retrieved via WebFetch within the time budget). This pass works from the canonical KN-I table of contents (Chs. I–VII + appendices, well-attested in every modern differential-geometry text), the peer-source crosswalks below, the Fast Track entry's editorial framing, and the in-Codex evidence that KN-I is already cited as the master anchor for 03.05.04, 03.05.07, 03.05.09, 03.06.05, 03.06.06 without a counterpart audit. A full P1 line-number inventory is deferred to the production pass when a local copy is on disk.

§1 What KN-I is for

KN-I is the canonical reference for the connection-and-curvature half of differential geometry. Where Lee's Introduction to Smooth Manifolds covers the manifold layer (charts, tangent bundles, vector fields, Lie derivative, integration) and Spivak's Comprehensive Introduction Vol. I walks the same material at a slower pace with more pictures, KN-I is the text that organises connections on principal $G$ -bundles, the Cartan structural equations for curvature and torsion, the holonomy group and its reduction theorems, and the specialisation to the affine and Riemannian cases. Every later text on gauge theory, geometric analysis, or Riemannian geometry either cites KN-I or rewrites its chapters in lighter notation [ref: Tu Differential Geometry: Connections, Curvature, and Characteristic Classes (GTM 275, Springer 2017) Preface ("the book is largely an exposition of the relevant chapters of Kobayashi-Nomizu Vol. I in modern notation"); Bleecker Gauge Theory and Variational Principles (Addison-Wesley 1981, Dover reprint 2005) Ch. 1 ("our principal reference for connections is Kobayashi-Nomizu Vol. I")].

Distinctive content, organised by the seven chapters of the book:

Chapter I — Differentiable manifolds. Charts and atlases, tangent and cotangent bundles, vector fields and the Lie bracket, the Frobenius theorem (involutive distributions integrate to foliations), Lie groups and Lie algebras as $G ⇝ g = T_{e} G$ , invariant vector fields, the exponential map $exp : g \to G$ , group actions (left, right, free, transitive, proper), and fibre bundles in general. The chapter sets up the entire infrastructure that the rest of the book depends on. Compare Lee Introduction to Smooth Manifolds Chs. 1–9 (verbose) and Spivak Vol. I Chs. 1–6 (geometric) [ref: Lee, Introduction to Smooth Manifolds (GTM 218, Springer 2nd ed. 2013) Chs. 1–9; Spivak, A Comprehensive Introduction to Differential Geometry Vol. I (Publish or Perish 3rd ed. 1999) Chs. 1–6].
Chapter II — Theory of connections. The technical heart of the book. Principal $G$ -bundle $P \to M$ with right $G$ -action; vertical subbundle $V P \subseteq T P$ ; a connection is a smooth right-equivariant horizontal complement $H P$ to $V P$ , equivalently a $g$ -valued connection 1-form $ω \in Ω^{1} (P; g)$ with $ω ∣_{V P} = id$ (after identification) and $R_{g}^{*} ω = Ad (g^{- 1}) ω$ . Horizontal lift of a curve in $M$ to $P$ ; parallel transport; the curvature 2-form $\Omega = \mathrm{d}\omega + \tfrac{1}{2}[\omega \wedge \omega]$ (Cartan's structural equation); the Bianchi identity $d_{ω} Ω = 0$ . Reduction of a principal $G$ -bundle to a subgroup $H \subseteq G$ and the resulting reduction of the connection. Associated bundles $E = P \times_{G} F$ , induced connections on $E$ , and the equivalence between connections on $P$ and covariant derivatives $\nabla$ on associated vector bundles (the bridge to the vector-bundle viewpoint). Compare Tu Differential Geometry Chs. 10–13 and Bleecker §3 [ref: Tu Chs. 10–13; Bleecker §3].
Chapter III — Linear and affine connections. Specialisation of Ch. II to the frame bundle $L (M) = Fr (M)$ , a principal $GL (n, R)$ -bundle. A linear connection on $M$ is a connection on $L (M)$ ; equivalently a covariant derivative on $T M$ . Affine connection: a linear connection together with a soldering form (the canonical $R^{n}$ -valued 1-form on $L (M)$ ). The torsion tensor $T (X, Y) = \nabla_{X} Y - \nabla_{Y} X - [X, Y]$ and the curvature tensor $R(X, Y) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$ (with the two Cartan structural equations $d θ = - ω \land θ + Θ$ and $d ω = - ω \land ω + Ω$ , where $Θ$ is the torsion form). Geodesics as auto-parallel curves of an affine connection; the exponential map $exp_{p}$ at a point. Normal coordinates around $p$ . Geodesic completeness.
Chapter IV — Riemannian connections. The unique Levi-Civita connection (1917): the unique torsion-free metric-compatible affine connection on a Riemannian manifold $(M, g)$ . Sectional curvature, the Ricci tensor, the scalar curvature. Hopf-Rinow theorem (geodesic completeness $\Leftrightarrow$ Cauchy completeness $\Leftrightarrow$ closed-bounded-compact). Spaces of constant curvature (Riemannian space forms): the model classification into $S^{n}$ , $R^{n}$ , $H^{n}$ at each curvature value, and the Killing-Hopf theorem that complete simply-connected constant- curvature spaces are exactly these. Compare Lee Riemannian Manifolds and do Carmo Riemannian Geometry [ref: Lee, Introduction to Riemannian Manifolds (GTM 176, Springer 2nd ed. 2018); do Carmo, Riemannian Geometry (Birkhäuser 1992)].
Chapter V — Curvature and space forms. Deeper development of constant-curvature spaces, classification, and the isometry group.
Chapter VI — Transformations. Affine and isometric transformations, infinitesimal isometries (Killing fields), the Myers-Steenrod theorem (the isometry group of a Riemannian manifold is a finite-dimensional Lie group), and transformations of connections.
Chapter VII — Holonomy. The holonomy group $\mathrm{Hol}(\omega, u) \subseteq G $o f a co nn ec t i o na t aba se p o in t$ u \in P$ — the subgroup generated by parallel transport around loops. Restricted holonomy $Hol^{0} (ω, u)$ via contractible loops. The Ambrose-Singer holonomy theorem (1953): the Lie algebra of $Hol^{0}$ is generated by the curvature $Ω_{u} (X, Y)$ as $u$ ranges over the holonomy bundle. The holonomy reduction theorem: a connection on $P$ reduces to a principal $Hol (ω, u)$ -bundle. Berger's classification (1955) of Riemannian holonomy is stated and referenced but the proof is left to the original literature; KN-II §X.5 expands on this. Compare Joyce Riemannian Holonomy Groups Ch. 2 [ref: Joyce Riemannian Holonomy Groups and Calibrated Geometry (Oxford Graduate Texts in Mathematics 12, OUP 2007) Ch. 2].
Editorial signature. KN-I is uncompromisingly principal-bundle first. Connections are defined on $P$ , not on $T M$ ; the vector-bundle / covariant-derivative viewpoint is recovered from associated bundles. This is the opposite of the order in Lee or do Carmo (which start from $\nabla$ on $T M$ ) and is the reason KN-I is the gauge-theory anchor: Yang-Mills requires the principal-bundle framing because the gauge group is the bundle automorphism group of $P$ , not the tangent bundle. Bleecker's textbook (cited above) is the bridge from KN's principal-bundle calculus to physical gauge theory [ref: Bleecker §3 ("we have adopted the Kobayashi-Nomizu principal- bundle formulation throughout because it is forced on us by physical gauge transformations")].

KN-I is not a first textbook on smooth manifolds. It assumes Chapter I material as a rapid review and pivots immediately to connections. The canonical "before KN-I" sequence is Lee Smooth Manifolds → Lee Riemannian Manifolds → KN-I, or Spivak Vol. I → KN-I. The canonical "after KN-I" sequence is KN-II (complex geometry, Chern-Weil, characteristic classes) → Bleecker / Donaldson-Kronheimer (gauge theory) → Joyce (special-holonomy / calibrated geometry).

§2 Coverage table (Codex vs KN-I)

Cross-referenced against the current Codex corpus. ✓ = covered, △ = partial / different framing, ✗ = not covered. KN-I material maps primarily to 03-modern-geometry/02-manifolds/, 03-lie/, 05-bundles/, 07-gauge-theory/, and adjacent.

KN-I topic	Codex unit(s)	Status	Note
Ch. I — manifold prerequisites
Smooth manifold, atlas, charts	`03.02.01-smooth-manifold.md`	✓	Single Codex manifold unit. Anchored on Lee. KN-I Ch. I §1 covered.
Tangent / cotangent bundle, vector fields	—	✗	Gap. Currently implicit in `03.02.01` and `03.05.02-vector-bundle.md`; no dedicated unit for $T M$ as a smooth vector bundle with the standard frame. Per the Milnor audit, `02-manifolds/` is "1 unit only — wide open"; this is one of the open slots.
Lie bracket $[X, Y]$ of vector fields, Lie derivative	—	✗	Gap. Foundational; no unit.
Frobenius theorem (involutive ⇒ integrable)	—	✗	Gap. Cited downstream (e.g. foliation references in `03.09-spin-geometry/`) but no anchor unit.
Lie group, Lie algebra	`03.03.01-lie-group.md`, `03.04.01-lie-algebra.md`	✓	Both shipped.
Group action (free, transitive, proper)	`03.03.02-group-action.md`	✓	Shipped.
Exponential map $exp : g \to G$ , Maurer-Cartan form	—	✗	Gap. Strange omission; cited by `03.05.07` and `03.05.09` without anchor.
Fibre bundle (general)	△	△	`03.05.01-principal-bundle.md`, `03.05.02-vector-bundle.md`, `03.05.10-sphere-bundle.md` cover the specialised cases. A general fibre-bundle unit (definition, local triviality, structure group, transition functions) is absent.
Ch. II — connections on principal bundles
Principal $G$ -bundle	`03.05.01-principal-bundle.md`	✓	Shipped.
Vertical subbundle $V P$ , fundamental vector fields	—	✗	Gap. Needed for the intrinsic definition of a connection 1-form. The existing `03.05.07` has it as an unanchored prerequisite (`lean_mathlib_gap` block flags "vertical tangent bundles, fundamental vector fields" as missing).
Connection on a principal bundle (horizontal distribution + connection 1-form)	`03.05.07-principal-bundle-connection.md`	△	Shipped, master anchor is "Kobayashi-Nomizu Vol. I §II". Adequate at FT-equivalence but the unit's `references` block has `source: TODO_REF` for the KN-I citation — the audit pass should resolve that.
Horizontal lift, parallel transport	—	✗	Gap. Currently a sub-topic inside `03.05.07` but the explicit horizontal-lift theorem (every curve in $M$ lifts uniquely to a horizontal curve in $P$ given an initial point) and the parallel-transport functor have no dedicated unit.
Curvature 2-form $Ω$ , Cartan structural equation $Ω = d ω + \frac{1}{2} [ω \land ω]$	`03.05.09-curvature.md`	✓	Shipped, master anchor is KN-I §III.
Bianchi identity $d_{ω} Ω = 0$	△	△	Mentioned inside `03.05.09` but no dedicated theorem statement; the algebraic Bianchi (first Bianchi $R (X, Y) Z + R (Y, Z) X + R (Z, X) Y = 0$ for torsion-free connections) is also absent.
Reduction of structure group; reduction of connection	—	✗	Gap. `03.05.03-orthogonal-frame-bundle.md` is the canonical example (reduction from $GL$ to $O$ via a metric) but the general reduction theorem is absent.
Associated bundle $E = P \times_{G} F$ ; induced connection	△	△	`03.05.02-vector-bundle.md` and `03.05.04-vector-bundle-connection.md` cover the vector-bundle specialisation. The general associated-bundle construction and the connection-induction theorem (KN-I §II.6) are not anchored.
Equivalence: connections on $P$ ↔ covariant derivatives on associated vector bundles	△	△	Implicit in the sibling pair `03.05.04` ↔ `03.05.07` but not stated as a theorem unit.
Ch. III — linear and affine connections
Frame bundle $L (M) = Fr (M)$	`03.05.03-orthogonal-frame-bundle.md`	△	The orthogonal frame bundle is shipped. The general $GL (n, R)$ -frame bundle (without a metric reduction) has no dedicated unit.
Linear connection on $M$ (= connection on $L (M)$ )	△	△	`03.05.04-vector-bundle-connection.md` covers the covariant-derivative side; the KN-I framing as a principal-bundle connection on $L (M)$ is not the entry point.
Soldering form / canonical $R^{n}$ -valued 1-form on $L (M)$	—	✗	Gap. Distinctive to KN-I; nothing analogous in Lee.
Torsion tensor $T (X, Y)$ and torsion 2-form $Θ$	—	✗	Gap. Foundational; no unit. Referenced informally in `03.05.09` as the second Cartan equation but never anchored.
Two Cartan structural equations	△	△	The curvature equation is in `03.05.09`; the torsion equation is missing.
Geodesic of an affine connection; auto-parallel curve	—	✗	Gap. Hopf-Rinow / Levi-Civita-shaped gap; flagged in the Milnor audit punch-list item `03.03.X1` (Levi-Civita / exp / Hopf-Rinow).
Exponential map $exp_{p}$ at a point; normal coordinates	—	✗	Gap. Cited downstream (Cartan-Hadamard, Bonnet-Myers) without anchor.
Geodesic completeness	—	✗	Gap.
Ch. IV — Riemannian connections
Levi-Civita connection (existence + uniqueness)	—	✗	Gap. This is the most-cited missing unit in the Codex: `03.09.18-berger-holonomy.md` invokes it without anchor; the Milnor audit lists it as priority-2 item `03.03.X1`. KN-I §IV is the canonical anchor.
Sectional curvature	—	✗	Gap.
Ricci tensor, scalar curvature	—	✗	Gap.
Hopf-Rinow theorem	—	✗	Gap. Shared with the Milnor audit punch-list.
Spaces of constant curvature; Riemannian space forms	—	✗	Gap.
Killing-Hopf theorem (simply-connected complete constant curvature = $S^{n}$ / $R^{n}$ / $H^{n}$ )	—	✗	Gap.
Ch. V–VI — transformations
Killing field, infinitesimal isometry	—	✗	Gap. Cited in symplectic and Lie-group units informally.
Myers-Steenrod (isometry group is a Lie group)	—	✗	Gap.
Ch. VII — holonomy
Holonomy group $Hol (ω, u)$ ; restricted holonomy $Hol^{0}$	△	△	`03.09.18-berger-holonomy.md` is shipped and uses the holonomy group as its central object, but it imports it without a definitional anchor unit. The Berger classification unit therefore has a load-bearing prerequisite gap.
Ambrose-Singer theorem (Lie algebra of $Hol^{0}$ = curvature span)	—	✗	Gap. Foundational holonomy theorem; absent from the Codex.
Holonomy reduction theorem	—	✗	Gap.
Berger's classification of Riemannian holonomy	`03.09.18-berger-holonomy.md`	△	Shipped, but as a downstream unit in the spin-geometry chapter; the upstream KN-I Ch. VII apparatus (Ambrose-Singer, restricted holonomy, reduction) is absent.
Ancillary / gauge-theory bridge
Yang-Mills action $∣ F_{A} ∣^{2}$ on a Riemannian base	`03.07.05-yang-mills-action.md`	✓	Shipped, but `07-gauge-theory/` contains only this one unit; the entire connection-theoretic apparatus it depends on is in `05-bundles/`. KN-I's Ch. II–III + the §VII holonomy material is the prerequisite chain for gauge theory.
Chern-Weil homomorphism	`03.06.06-chern-weil-homomorphism.md`	✓	Shipped, master anchor cites KN-II (complex characteristic classes), but the construction itself (invariant polynomial of curvature) uses KN-I §III.
Invariant polynomial	`03.06.05-invariant-polynomial.md`	✓	Shipped.
Pontryagin and Chern classes	`03.06.04-pontryagin-chern-classes.md`	✓	Shipped.

Aggregate coverage estimate: ~20–25% of KN-I has corresponding Codex units. Chapter II (connections on principal bundles) is the best-covered chapter at ~60%, because 03.05.01, 03.05.04, 03.05.07, 03.05.09 together cover the principal-bundle-connection-and-curvature core (though with the gaps flagged above: horizontal lift, parallel transport, reduction, associated bundles, torsion). Chapter III (linear/affine connections) is ~15% covered: only the orthogonal frame bundle and the covariant- derivative side, with no soldering form, no torsion tensor, no geodesics as auto-parallels, no exp map, no normal coordinates. Chapter IV (Riemannian connections) is ~0% covered: Levi-Civita itself is absent as a unit, and Hopf-Rinow, sectional / Ricci / scalar curvature, and the space forms are all missing. Chapters V–VI are essentially uncovered. Chapter VII (holonomy) has the downstream classification unit (03.09.18) but the upstream apparatus (Ambrose-Singer, restricted holonomy, holonomy reduction) is missing.

Silent KN-I dependencies in the Codex. The audit reveals a cluster of units that cite KN-I as their master anchor but whose load- bearing prerequisites are uncovered:

03.05.04-vector-bundle-connection.md (master = "Kobayashi-Nomizu Vol. I §III") — silently depends on a missing general fibre-bundle unit and a missing Lie derivative / vector-field unit.
03.05.07-principal-bundle-connection.md (master = "Kobayashi-Nomizu Vol. I §II; Steenrod §17") — silently depends on a missing vertical subbundle and fundamental vector field unit, and the unit's own lean_mathlib_gap block already flags these as the formalisation blockers.
03.05.09-curvature.md (master = "Kobayashi-Nomizu Vol. I §III") — states the curvature Cartan equation but not the torsion equation; silently depends on a missing torsion tensor unit and a missing Bianchi identity unit.
03.05.03-orthogonal-frame-bundle.md — is the example of a reduction of structure group from $GL (n)$ to $O (n)$ without an anchor unit for the general reduction theorem.
03.07.05-yang-mills-action.md — the entire gauge-theory chapter is one unit whose Master tier cites Atiyah-Bott and Donaldson-Kronheimer; both anchor texts assume KN-I Ch. II–III + Ch. VII as prerequisites that the Codex does not supply at present.
03.09.18-berger-holonomy.md — cites Berger 1955 directly, bypasses KN-I Ch. VII (Ambrose-Singer / holonomy reduction). The definition of the holonomy group is invoked without an anchor.
03.06.06-chern-weil-homomorphism.md — Chern-Weil is the application of KN-I Ch. III curvature; the apparatus is partly present (03.05.09 + 03.06.05) but the invariant-polynomial-of- curvature construction depends on the torsion-free / structural- equation packaging that the torsion gap above weakens.

Net effect: the existing Codex 05-bundles/ chapter cites KN-I as its canonical reference but covers ~60% of the relevant material; the 02-manifolds/ chapter (the prerequisite layer) covers ~10%; the 07-gauge-theory/ chapter (the downstream layer) cites KN-I via intermediaries but has no other units. The KN-I audit is therefore load-bearing across three chapters.

§3 Gap punch-list (P3-lite — units to write, priority-ordered)

Priority 0 — manifold-layer prerequisites (also on the Milnor / Lee-equivalent open punch-list for 02-manifolds/):

These are shared with any future Lee Introduction to Smooth Manifolds audit and with the Milnor Topology from the Differentiable Viewpoint audit. Listed here because KN-I assumes them and the audit reveals them as silently load-bearing.

03.02.02 Tangent bundle as a smooth vector bundle. $T M$ with its canonical smooth structure, $π : T M \to M$ , local frames in a chart. KN-I §I.2 anchor; Lee Ch. 3 anchor. Three-tier, ~1500 words.
03.02.03 Vector fields, Lie bracket, Lie derivative. $Γ (T M)$ , the Lie bracket as a Lie-algebra structure on $X (M)$ , Lie derivative $L_{X}$ on tensor fields. KN-I §I.3 anchor; Lee Chs. 8–12 anchor. ~1500 words. Three-tier.
03.02.04 Frobenius theorem. Involutive ⇔ integrable distribution. KN-I §I.2 (Theorem 2.1) anchor; Lee Ch. 19 anchor. ~1200 words. Intermediate + Master.
03.03.04 Exponential map $exp : g \to G$ and the Maurer-Cartan form. KN-I §I.4 anchor; Warner Ch. 3 anchor. Three-tier, ~1500 words. Sits in the existing 03-lie/ chapter.
03.05.00 General fibre bundle (definition, local triviality, structure group, transition functions). The umbrella unit that sits above 03.05.01-principal-bundle.md and 03.05.02-vector-bundle.md. KN-I §I.5 anchor; Steenrod Topology of Fibre Bundles §2–§3 anchor. ~1500 words. Three-tier.

Priority 1 — load-bearing KN-I Ch. II–III core (the heart of the audit):

03.05.06 Vertical subbundle and fundamental vector fields. Inside 05-bundles/. $V P = ker (d π) \subseteq T P$ ; for each $X \in g$ the fundamental vector field $X^{*}$ on $P$ . The infinitesimal-action map $g \to Γ (V P)$ is an isomorphism onto vertical fields. KN-I §II.1 anchor. ~1200 words. Intermediate + Master. Unblocks the 03.05.07 Lean formalisation.
03.05.11 Horizontal lift and parallel transport. Existence and uniqueness of horizontal lifts of curves in $M$ to $P$ . The parallel-transport functor $\mathrm{P}\gamma: P{\gamma(0)} \to P_{\gamma(1)} $a s a$ G$-equivariant map. KN-I §II.3 anchor. ~1500 words. Three-tier.
03.05.12 Reduction of structure group; reduction of a connection. General theorem: a section of $P / H \to M$ ⇔ a reduction of $P$ to an $H$ -subbundle $Q \subseteq P$ . A connection $ω$ reduces to $Q$ iff its holonomy at a point of $Q$ lies in $H$ . KN-I §II.7 anchor. ~1500 words. Three-tier. The orthogonal-frame-bundle unit 03.05.03 becomes a canonical example of this theorem.
03.05.13 Associated bundle and induced connection. $E = P \times_{G} F$ for $G$ acting on $F$ ; the connection on $P$ induces a connection on $E$ (and a covariant derivative when $F$ is a vector space). KN-I §II.6 anchor. ~1500 words. Three-tier. The pair 03.05.04 ↔ 03.05.07 becomes a corollary of this.
03.05.14 Torsion tensor and the two Cartan structural equations. Torsion $T (X, Y) = \nabla_{X} Y - \nabla_{Y} X - [X, Y]$ , the torsion 2-form $Θ = d θ + ω \land θ$ on $L (M)$ , the full pair of Cartan equations. The first and second Bianchi identities. KN-I §III.2, §III.5 anchor. ~1500 words. Three-tier. Closes the 03.05.09-curvature.md partial coverage.
03.05.15 Linear connection via the frame bundle; soldering form. The canonical $R^{n}$ -valued 1-form $θ$ on $L (M)$ (soldering form), and the resulting structural equations. KN-I §III.2 anchor. ~1500 words. Master-tier dominant; the Beginner tier is the Lee-style " $\nabla$ on $T M$ " reformulation.
03.03.X1 Levi-Civita connection, geodesics, exponential map, Hopf-Rinow. Shared with the Milnor audit (milnor-morse-theory.md punch-list item 6). KN-I §IV.1–§IV.4 anchor; do Carmo Riemannian Geometry Ch. 2–7 secondary anchor. ~2500 words. Three-tier. Single most-cited missing unit in the Codex.

Priority 2 — Ch. IV–V Riemannian completions and Ch. VII holonomy:

03.02.05 Sectional curvature, Ricci tensor, scalar curvature. The three standard curvature contractions. KN-I §IV.1 anchor; Lee Riemannian Manifolds Ch. 8 secondary. ~1500 words. Three-tier.
03.02.06 Constant-curvature spaces and Killing-Hopf. Model spaces $S^{n}$ , $R^{n}$ , $H^{n}$ ; classification of complete simply-connected constant-curvature manifolds. KN-I §V.3 anchor. ~1500 words. Three-tier.
03.05.16 Holonomy group and restricted holonomy. $Hol (ω, u)$ and $Hol^{0} (ω, u)$ as Lie subgroups of $G$ . The holonomy bundle. KN-I §VII.1–§VII.2 anchor. ~1500 words. Three-tier. Supplies the missing prerequisite for 03.09.18-berger-holonomy.md.
03.05.17 Ambrose-Singer holonomy theorem. The Lie algebra of $Hol^{0}$ equals the span of $Ω_{u} (X, Y)$ as $u$ ranges over the holonomy bundle and $X, Y \in T_{u} P$ . KN-I §VII.8 anchor; Ambrose-Singer 1953 originator. ~1500 words. Intermediate + Master.
03.05.18 Holonomy reduction theorem. A connection on $P$ reduces to a principal $Hol (ω, u)$ -bundle. KN-I §VII.2 anchor. ~1200 words. Master-dominant.

Priority 3 — Ch. VI transformations + Bianchi packaging:

03.02.07 Killing fields and infinitesimal isometries. Killing equation $L_{X} g = 0$ ; the Lie algebra of Killing fields. KN-I §VI.2 anchor. ~1200 words. Intermediate + Master.
03.02.08 Myers-Steenrod theorem. The isometry group of a Riemannian manifold is a finite-dimensional Lie group. KN-I §VI.3 anchor. ~1200 words. Master-dominant.
First and second Bianchi identities — add as a section to 03.05.14 rather than a new unit.

Priority 4 — survey pointers (optional, Master-only):

Pointer in 03.07.05-yang-mills-action.md to the new 03.05.11–03.05.18 chain as the prerequisite anchor. Single- paragraph weaving edit, not a new unit; recorded here so it is not forgotten in Pass-W.
Pointer in 03.09.18-berger-holonomy.md to 03.05.16–03.05.18 as the upstream holonomy apparatus. Single-paragraph weaving edit.
Pointer in 03.06.06-chern-weil-homomorphism.md to 03.05.14 (torsion-free, structural equations) — the construction is sharpened once the torsion apparatus is anchored. Single-paragraph weaving edit.

§4 Implementation sketch (P3 → P4)

For a full KN-I coverage pass, priority-0+1 are the minimum set (12 units) and close the gauge-theory and holonomy silent-dependency gaps. Realistic production estimate (mirroring earlier Brown / Bott-Tu / Milnor batches):

~3 hours per unit. KN-I units skew slightly above the corpus average because the Master tier requires careful principal-bundle notation (vertical / horizontal / fundamental / equivariant) and the structural equations need both the moving-frame and the abstract presentation.
Priority 0: 5 units × ~2.5 hours = ~12 hours. Mostly manifold-layer patches with strong external anchors (Lee).
Priority 1: 7 units × ~3.5 hours = ~25 hours. Heart of the audit; closes the gauge-theory and Riemannian-connection gaps in one batch.
Priority 2: 5 units × ~3 hours = ~15 hours. Closes the holonomy chain into 03.09.18.
Priority 3: 2 units + 1 section × ~2 hours = ~5 hours.
Priority 4: 3 weaving edits × ~30 min = ~1.5 hours.
Total for full coverage: ~58 hours, roughly two focused weeks. Priority-0+1 alone is ~37 hours and closes the load-bearing gaps.

Originator-prose targets. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, the following units should carry originator-prose citations:

Tullio Levi-Civita, "Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana," Rendiconti del Circolo Matematico di Palermo 42 (1917) 173–204 — the original definition of the Levi-Civita connection on a Riemannian manifold. Cite in 03.03.X1.
Élie Cartan, "Sur les variétés à connexion affine et la théorie de la relativité généralisée," Annales scientifiques de l'É.N.S. 40 (1923) 325–412, cont'd 41 (1924) 1–25, 42 (1925) 17–88 — the invention of the moving-frame approach to connections; the Cartan structural equations are introduced here. Cite in 03.05.14 and 03.05.15.
Charles Ehresmann, "Les connexions infinitésimales dans un espace fibré différentiable," in Colloque de topologie de Bruxelles (1950) — the formal definition of a connection on a principal bundle as a horizontal distribution. Cite in 03.05.07 and 03.05.11.
Warren Ambrose, Isadore Singer, "A theorem on holonomy," Trans. AMS 75 (1953) 428–443 — the holonomy theorem. Cite in 03.05.17.
Marcel Berger, "Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes," Bull. Soc. Math. France 83 (1955) 279–330 — the holonomy classification. Already cited in 03.09.18-berger-holonomy.md; the new upstream units (03.05.16–03.05.18) should cross-reference.
Heinz Hopf, Willi Rinow, "Über den Begriff der vollständigen differentialgeometrischen Fläche," Comment. Math. Helv. 3 (1931) 209–225 — Hopf-Rinow. Cite in 03.03.X1.
Sumner Myers, Norman Steenrod, "The group of isometries of a Riemannian manifold," Annals of Math. 40 (1939) 400–416 — Myers-Steenrod. Cite in 03.02.08.
Kobayashi, Nomizu (1963) — the canonical consolidation. Cite throughout as the in-Codex master anchor.

Notation crosswalk. KN-I writes:

$P (M, G)$ for a principal $G$ -bundle over $M$ (instead of $P \to M$ ).
$ω$ for the connection 1-form, $Ω$ for the curvature 2-form.
$Θ$ for the torsion 2-form on $L (M)$ , $θ$ for the soldering / canonical $R^{n}$ -valued 1-form.
$L (M) = Fr (M)$ for the frame bundle.
$Hol (ω, u)$ and $Hol^{0} (ω, u)$ for full and restricted holonomy at $u \in P$ .
$T (X, Y)$ , $R (X, Y) Z$ for torsion and curvature as tensors (alongside $Ω, Θ$ as forms).

Tu Differential Geometry uses identical notation; Bleecker writes $F = F_{A}$ for curvature (the gauge-theory convention) and $A$ for $ω$ (gauge potential). Lee writes $\nabla$ throughout and introduces the principal-bundle viewpoint only in Ch. 17. The Codex notation decision (per docs/specs/UNIT_SPEC.md §11) should: adopt KN-I's $ω$ , $Ω$ , $Θ$ , $θ$ , $L (M)$ verbatim for the principal-bundle units; also introduce the gauge-theory $A = ω$ , $F = Ω$ aliases in 03.07.05-yang-mills-action.md and in 03.05.11 (horizontal lift / parallel transport), with a §Notation paragraph cross-referencing both.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem in KN-I (full P1 audit). Deferred until a local PDF is on disk in reference/fasttrack-texts/03-modern-geometry/Kobayashi-Nomizu-FoundationsVol1.pdf. This stub works from canonical TOC knowledge + Codex internal evidence and is therefore REDUCED.
Kobayashi-Nomizu Foundations of Differential Geometry, Vol. II (Wiley 1969) — Fast Track 3.19. Deferred to a separate audit. Vol. II covers: complex and almost-complex structures, Hermitian and Kähler manifolds, characteristic classes via Chern-Weil (overlapping with Codex 03.06-characteristic-classes/), homogeneous spaces, and submanifolds. The two volumes are typically cited together but are substantively distinct chapters of differential geometry.
Yang-Mills theory specifics (instanton moduli spaces, Donaldson invariants, the Atiyah-Bott moment-map picture, BPST instantons) — Fast Track 3.20 (Donaldson-Kronheimer / Atiyah-Bott). Deferred. The 03.07.05-yang-mills-action.md unit already exists as the entry point; the deeper Yang-Mills audit is its own pass.
Riemannian comparison geometry (Toponogov, volume comparison, splitting theorems) — KN-I touches the Bonnet-Myers / Cartan-Hadamard end only; the full comparison-geometry programme is do Carmo Riemannian Geometry / Petersen Riemannian Geometry, not KN-I. Deferred.
Geometric analysis / Ricci flow — far downstream of KN-I. Deferred indefinitely.
Foliations and $G$ -structures beyond reduction of structure group — KN-II §I covers $G$ -structures more thoroughly. Deferred to the KN-II audit.

§6 Acceptance criteria for FT equivalence (KN-I)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, the book is at equivalence-coverage when:

≥95% of KN-I's named theorems in Chs. I–VII map to Codex units. Current ~20–25%; after priority-0 patches rises to ~40%; after priority-0+1 to ~75%; after priority-0+1+2 to ~92%; full ≥95% requires priority-3 (priority-4 is weaving, not new units).
≥90% of KN-I's worked examples (the canonical principal-bundle examples — $Fr (M)$ , $O (M)$ , the Hopf bundle $S^{3} \to S^{2}$ , Stiefel manifolds, symmetric spaces $G / H$ ) have either a direct unit or are referenced from a unit covering them.
All five units in 03.05-bundles/ that currently list a source: TODO_REF for the KN-I citation (03.05.04, 03.05.07, 03.05.09, and via aliases 03.05.03, 03.05.01) have the TODO_REF resolved to a concrete chapter/section locator in KN-I.
03.09.18-berger-holonomy.md cites the new upstream holonomy units (03.05.16–03.05.18) as prerequisites.
03.07.05-yang-mills-action.md cites the new connection-reduction and parallel-transport units (03.05.11–03.05.13) as prerequisites via Pass-W weaving paragraphs.
The Milnor-audit shared item (03.03.X1 Levi-Civita / Hopf-Rinow) ships once and is cited by both audits without duplication.
Notation decisions are recorded in 03.05.07, 03.05.11, 03.05.14, 03.05.15 (see §4).
Originator-prose citations of Levi-Civita 1917, Cartan 1923–25, Ehresmann 1950, Ambrose-Singer 1953, Hopf-Rinow 1931, and Myers-Steenrod 1939 are present in the relevant units.

The 5 priority-0 + 7 priority-1 units close the gauge-theory and Riemannian-connection silent-dependency gaps and lift KN-I coverage from ~25% to ~75%. Priority-2 closes the holonomy chain (the most important remaining gap because it is load-bearing for the existing 03.09.18-berger-holonomy.md). Priority-3 closes the Ch. VI transformations chapter and Bianchi packaging.

§7 Sourcing

Not free. KN-I is under active Wiley copyright (1963, reprint 1996). No free legal PDF is hosted by the authors' institutions or by Wiley. Print copies (Wiley Classics Library reprint, ISBN 0-471-15733-3) are readily available second-hand.
Local copy. Not present in reference/textbooks-extra/ or reference/fasttrack-texts/03-modern-geometry/ at the time of this audit. The Fast Track entry holds only a cover-image stub (reference/fast-track/images/Kobayashi-Nomizu-1-683x1024__72960fe9e3.jpg). A scanned copy should be added to reference/fasttrack-texts/03-modern-geometry/Kobayashi-Nomizu-FoundationsVol1.pdf before the production pass on the priority-1 punch-list units, so that line-number citations can be resolved in the TODO_REF blocks of 03.05.04, 03.05.07, 03.05.09.
Companion peer texts (cited in §1):
- L. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, Graduate Texts in Mathematics 275, Springer 2017. The most accessible modern rewriting of KN-I; explicitly designed as a teaching version of KN-I + KN-II Ch. XII. The preferred secondary anchor for priority-1 units.
- M. Spivak, A Comprehensive Introduction to Differential Geometry, Vols. I–II, Publish or Perish (3rd ed. 1999). The most geometric and historical treatment; Vol. II Ch. 4–8 parallels KN-I Ch. II–IV with extensive figures and historical notes (Riemann's Habilitationsschrift, Levi-Civita's discovery of parallelism, Cartan's moving frames).
- D. Bleecker, Gauge Theory and Variational Principles, Addison-Wesley 1981, Dover reprint 2005. The bridge from KN-I's principal-bundle calculus to physical gauge theory; Ch. 3 is the most accessible exposition of why physicists need KN-I and not Lee. Cite in 03.07.05-yang-mills-action.md Pass-W weaving.
- J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, Springer (2nd ed. 2013). The standard prerequisite. Anchor for the priority-0 manifold-layer units (03.02.02, 03.02.03, 03.02.04).
- J. M. Lee, Introduction to Riemannian Manifolds, GTM 176, Springer (2nd ed. 2018). The standard Riemannian-geometry text; anchor for the priority-2 Riemannian units (03.02.05, 03.02.06) alongside KN-I §IV.
- D. Joyce, Riemannian Holonomy Groups and Calibrated Geometry, Oxford Graduate Texts in Mathematics 12, OUP 2007. The standard reference for special holonomy beyond Berger's classification; cited in 03.09.18 and the new priority-2 holonomy units.
Originator-paper archive locations:
- Levi-Civita 1917 Rend. Circ. Mat. Palermo — open access via BDIM / Biblioteca Digitale Italiana di Matematica.
- Cartan 1923–25 Annales sci. ÉNS — open access via NUMDAM.
- Ehresmann 1950 Colloque de topologie de Bruxelles — collected works; print only.
- Ambrose-Singer 1953 Trans. AMS — JSTOR.
- Berger 1955 Bull. Soc. Math. France — open access via NUMDAM.
- Hopf-Rinow 1931 Comment. Math. Helv. — Springer archive (paywall).
- Myers-Steenrod 1939 Annals — JSTOR.

Kobayashi-Nomizu — *Foundations of Differential Geometry, Vol. I* (Fast Track 3.18) — Audit + Gap Plan