Sternberg — *Lectures on Differential Geometry* (Fast Track 1.10) — Audit + Gap Plan

Book: Shlomo Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs NJ 1964, xi + 390 pp. Reissued by Chelsea Publishing 1983 (ISBN 0-8284-0316-3); now distributed by AMS / Chelsea Series. Sternberg's first major textbook and the canonical 1960s American exposition of differential geometry in the Cartan / invariant-theoretic style.

Fast Track entry: 1.10. Distinct from Sternberg's later books on Curvature in Mathematics and Physics (FT 1.14, Dover 2012) and Group Theory and Physics (FT 1.15, CUP 1994) — those are separate audits and are out of scope here.

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 to write so that Lectures on Differential Geometry (LDG 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. The author's Harvard page (people.math.harvard.edu/~shlomo/) hosts six of Sternberg's books — Real Variables, Advanced Calculus, Dynamical Systems, Lie Algebras, Semi-Riemannian Geometry, Semi-Classical Analysis — but not Lectures on Differential Geometry, which remains under active Chelsea / AMS copyright. WebFetch on the Harvard page confirmed the omission. The five Sternberg PDFs already in reference/fasttrack-texts/01-fundamentals/ and 02-quantum-stat/ and 03-modern-geometry/ are the later Sternberg books (FT 1.04, 1.14, 1.15, 2.x adjacent); none is the 1964 LDG. A full P1 line-number inventory is deferred to the production pass when a local copy is on disk. This stub works from the canonical LDG table of contents (Chs. I–VII / VIII depending on edition, well-attested across modern differential-geometry references that cite LDG as an originator anchor — Cartan structural equations, $G$-structures, Cartan's prolongation algorithm, Spencer cohomology), the peer-source crosswalks below, and the in-Codex evidence that LDG is the most-cited 1960s source for the Cartan moving-frame side of differential geometry that KN-I does not emphasise.

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§1 What LDG is for

LDG is the Cartan-tradition counterpart to Kobayashi-Nomizu Vol. I. Where KN-I (FT 3.18, written almost simultaneously — KN-I 1963, LDG 1964) organises differential geometry around principal $G$-bundles and the connection 1-form $\omega \in {\Omega }^1(P;\mathfrak{g})$, Sternberg organises the same material around the moving frame $(e_1, \ldots, e_n)$ on the manifold itself together with its dual coframe $({\theta }^1,\dots ,{\theta }^n)$ and the resulting Cartan structural equations $\mathrm{d}{\theta }^i=-{\omega }_j^i\wedge {\theta }^j+{\Theta }^i$, $\mathrm{d}{\omega }_j^i=-{\omega }_k^i\wedge {\omega }_j^k+{\Omega }_j^i$. The two presentations are mathematically equivalent (a principal-bundle connection on the frame bundle $L(M)$ pulls back to a moving-frame connection on $M$ via a local section), but the editorial style is opposite: KN-I is uncompromisingly abstract and reads as a research monograph; LDG is uncompromisingly geometric, written to be read chapter-by-chapter, and is the canonical American 1960s differential- geometry course text. Tu's Differential Geometry: Connections, Curvature, and Characteristic Classes (GTM 275, Springer 2017) was written specifically to be "Sternberg in modern notation"; the Preface explicitly cites LDG as the source of the moving-frame examples [ref: Tu Differential Geometry GTM 275 Preface; Sternberg LDG was the textbook for Tu's course at Tufts in the 1980s].

Distinctive content, organised by the seven (or eight) chapters of the book:

1. Chapter I — The geometry of Euclidean space and the calculus of functions of several variables. Sternberg's signature opening: develops differential calculus on ${\mathbb{R}}^n$ with the coordinate-free / multilinear-algebra perspective from the outset. Inverse and implicit function theorems with the Banach-space proof. Sard's theorem (Sard 1942) is stated and proved early — unusual for the era; most 1960s texts deferred Sard to a later chapter. The chapter ends with the existence-uniqueness theorem for ODEs and the smooth-dependence-on-initial-conditions theorem, setting up flows of vector fields. Compare Spivak Vol. I Chs. 1–3 (more pictures, slower pace) and Lang Differential and Riemannian Manifolds (GTM 160, Springer 1995) Chs. I–IV (Banach-manifold from the outset, the most abstract route) [ref: Spivak, A Comprehensive Introduction to Differential Geometry Vol. I (Publish or Perish 3rd ed. 1999) Chs. 1–3; Lang, Differential and Riemannian Manifolds GTM 160 Springer 1995 Chs. I–IV].
2. Chapter II — Differentiable manifolds. Standard differentiable- manifold setup. Charts, atlases, smooth maps, submersions and immersions, the regular value theorem, partitions of unity, embedding theorems (Whitney 1936 — Sternberg gives the easy Whitney embedding $M^n\hookrightarrow {\mathbb{R}}^{2n+1}$, not the strong $M^n\hookrightarrow {\mathbb{R}}^{2n}$). Tangent and cotangent bundles introduced as derivations and as germs of functions. The chapter is short relative to LDG's chapter on Lie groups (Ch III) — Sternberg moves rapidly through manifold-layer foundations to get to the geometry that motivated the book.
3. Chapter III — Integration of vector fields. Lie groups and Lie algebras (early treatment). Sternberg's pedagogically distinctive choice: Lie groups are introduced before differential forms. Where KN-I introduces them as a special case of the bundle-and-connection formalism, and where Lee Introduction to Smooth Manifolds puts them late (Ch. 20+) after the manifold infrastructure is in place, LDG develops Lie groups in chapter III as the natural recipients of the flow of a vector field programme: a one-parameter subgroup $\mathbb{R}\to G$ is the integral curve of a left-invariant vector field. The exponential map $\exp :\mathfrak{g}\to G$ is then the time-1 flow. Cartan's theorem (closed subgroup of a Lie group is a Lie subgroup) is proved; the Lie algebra ↔ Lie group correspondence is developed up to Lie's third theorem (every finite-dim Lie algebra is the Lie algebra of a Lie group) for the simply-connected case. Frobenius theorem is proved here, as the integrability statement for distributions of vector fields. This ordering is distinctive of LDG and is preserved in only a few later texts (Warner, Foundations of Differentiable Manifolds and Lie Groups (GTM 94, Springer 1983) follows the same Lie-first ordering; do Carmo Riemannian Geometry and Lee do not) [ref: Warner, Foundations of Differentiable Manifolds and Lie Groups GTM 94, Springer 1983, Chs. 1–3].
4. Chapter IV — The integral calculus of forms (differential forms + exterior derivative + Stokes' theorem). Differential forms ${\Omega }^k(M)$ introduced via the exterior algebra ${\Lambda }^k{T}^{\ast }M$. The exterior derivative $\mathrm{d}:{\Omega }^k\to {\Omega }^{k+1}$ characterised by the four axioms ($\mathbb{R}$-linearity, Leibniz $\mathrm{d}(\alpha \wedge \beta) = \mathrm{d}\alpha \wedge \beta + (-1)^k \alpha \wedge \mathrm{d}\beta$,agreementwith$\mathrm{d}f$ on 0-forms, ${\mathrm{d}}^2=0$) — Sternberg gives the axiomatic uniqueness proof, not just the coordinate formula. Pullback $f^{\ast }\alpha$ and its naturality with $\mathrm{d}$. Integration ${\int }_M\alpha$ on oriented manifolds (top-dim forms). Stokes' theorem ${\int }_{\partial M}\alpha ={\int }_M\mathrm{d}\alpha$ for oriented manifolds-with-boundary; the proof is the partition-of-unity
   • chart-by-chart reduction to Stokes on ${\mathbb{R}}_{+}^n$. The chapter ends with de Rham cohomology $H_{\mathrm{dR}}^k(M)$ as $\ker \mathrm{d}/\mathrm{im}\mathrm{d}$ and a sketch of the de Rham theorem $H_{\mathrm{dR}}^k(M)\cong H^k(M;\mathbb{R})$ (without the full sheaf-theoretic proof). Compare Bott-Tu Differential Forms in Algebraic Topology (GTM 82, Springer 1982) Chs. I–II [ref: Bott, Tu, Differential Forms in Algebraic Topology GTM 82, Springer 1982].
5. Chapter V — Riemannian geometry, connections, and the Cartan structural equations. The chapter that gives LDG its lasting reputation. Riemannian metric $g$ on $M$. The Levi-Civita connection is constructed via the unique torsion-free compatible covariant derivative; Sternberg gives both the abstract construction (as in KN-I) and the moving-frame construction (orthonormal frame $(e_1,\dots ,e_n)$, dual coframe $({\theta }^i)$, connection 1-forms ${\omega }_j^i$ with ${\omega }_i^j=-{\omega }_j^i$ for $\mathrm{O}(n)$-reduction). First structural equation $\mathrm{d}{\theta }^i+{\omega }_j^i\wedge {\theta }^j=0$ (torsion-free). Second structural equation $\mathrm{d}\omega^i_j + \omega^i_k \wedge \omega^k_j = \Omega^i_j$ (curvature). Bianchi identities in both algebraic ($R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0$) and differential (${\mathrm{d}}_{\omega }\Omega =0$) form. Sectional, Ricci, scalar curvature. Geodesics as auto-parallels. Exponential map ${\exp }_p$ and normal coordinates. Hopf-Rinow theorem (geodesic completeness = Cauchy completeness = closed- bounded-compact). Where LDG diverges from KN-I: the moving-frame calculus is the default mode; the principal-bundle formalism is presented as a clean reformulation, not as the entry point.
6. Chapter VI — The geometry of $G$-structures. The chapter without which LDG is not LDG. A $G$-structure on $M$ is a reduction of the frame bundle $L(M)\to M$ to a principal $G$- subbundle, for $G\subseteq \mathrm{GL}(n,\mathbb{R})$. Examples: $\mathrm{O}(n)$-structure = Riemannian metric; $\mathrm{Sp}(2n)$- structure = almost-symplectic; $\mathrm{GL}(n,\mathbb{C})$ for $n$ even = almost-complex; volume / orientation = $\mathrm{SL}(n)$- structure. Integrability of a $G$-structure: when is there a coordinate atlas in which the structure is the flat model? Cartan's prolongation algorithm: build the first prolongation $G^{(1)} \subseteq \mathfrak{g} \otimes (\mathbb{R}^n)^*$ via the Spencer cohomology of the inclusion $\mathfrak{g} \hookrightarrow \mathfrak{gl}(n, \mathbb{R})$,iterate.A$G$-structure is of finite type if some prolongation vanishes; of infinite type otherwise. Riemannian and conformal $G$-structures are of finite type (so the local geometry is determined by finitely many jets); symplectic and complex are of infinite type (which is why symplectic geometry has no local invariants — Darboux's theorem — and complex geometry depends on holomorphic data). The classification of finite-type $G$-structures is the Sternberg / Guillemin / Singer theorem [ref: V. Guillemin, S. Sternberg, "An algebraic model of transitive differential geometry," Bull. AMS 70 (1964) 16–47]. This chapter is essentially absent from KN-I and Lee, partial in Spivak Vol. II, and present in modern form only in Sternberg's own later work and in selected research monographs (Yano-Kon, Kobayashi Transformation Groups in Differential Geometry, and Guillemin- Sternberg Geometric Asymptotics). It is the single most distinctive feature of LDG and the strongest argument for a dedicated Sternberg audit beyond the KN-I overlap.
7. Chapter VII — The classical mechanics of particles and rigid bodies. Symplectic structures, Hamiltonian flows, the principle of stationary action. Differential geometry applied to classical mechanics. Symplectic manifold $(M^{2n},\omega )$. Hamiltonian vector field $X_H$ defined by ${\iota }_{X_H}\omega =\mathrm{d}H$. Poisson bracket $\{f,g\}$ and the Lie-algebra structure on $C^{\infty }(M)$. Liouville's theorem (${\omega }^n$ preserved by Hamiltonian flow). Variational principles — Hamilton's principle of stationary action; the Euler-Lagrange equations as the first-order conditions; the Legendre transform $TM\to T^{\ast }M$. This chapter foreshadows Sternberg's later geometric-mechanics programme (Marsden-Weinstein reduction, moment maps, Guillemin- Sternberg convexity, eventually FT 1.14 Curvature in Math and Physics and FT 1.15 Group Theory and Physics). Compare Arnold Mathematical Methods of Classical Mechanics (GTM 60, Springer 2nd ed. 1989) Chs. 7–9 [ref: Arnold, Mathematical Methods of Classical Mechanics GTM 60, Springer 2nd ed. 1989, Chs. 7–9].
8. Editorial signature. Three threads make LDG distinctive against KN-I:
   • Lie groups before differential forms (Ch III before Ch IV). Pedagogically motivates the exterior algebra as a tool for describing Lie-algebra-valued forms.
   • Moving frames as the default mode (Ch V). Principal-bundle formalism is a reformulation, not the entry point.
   • $G$-structures and prolongation (Ch VI). Almost no overlap with KN-I; the invariant-theoretic / Cartan-Spencer programme is LDG-only at this level.

LDG is not a first textbook on smooth manifolds — Chapter I is a review, not a development. The canonical "before LDG" sequence is Sternberg's own Advanced Calculus (which IS free at the Harvard page) or Spivak Calculus on Manifolds → LDG. The canonical "after LDG" sequence is KN-I (for the principal-bundle viewpoint) → KN-II (Chern- Weil, characteristic classes) → Sternberg's own Curvature in Math and Physics (FT 1.14) for the gauge-theory / GR application layer.

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§2 Coverage table (Codex vs LDG)

Cross-referenced against the current Codex corpus. ✓ = covered, △ = partial / different framing, ✗ = not covered. LDG material maps primarily to 03-modern-geometry/02-manifolds/, 03-lie/, 04-differential-forms/, 05-bundles/, and 05-classical-mechanics/. Heavy overlap with the KN-I (FT 3.18) audit is flagged explicitly; shared punch-list items should ship once.

LDG topic	Codex unit(s)	Status	Note (overlap with KN-I audit)
Ch. I — calculus on ${\mathbb{R}}^n$ and ODE existence
Inverse / implicit function theorem	—	✗	Gap. Standard prerequisite; on the Apostol-multivariable audit.
Sard's theorem	—	✗	Gap. Cited downstream (Milnor MMT audit, transversality) without anchor.
Existence-uniqueness for ODEs; smooth dependence	△	△	Concept-catalog entry exists; no dedicated unit. Shared with Arnold-ODE audit.
Flow of a vector field	△	△	Concept-catalog entry; no dedicated unit. Shared with KN-I priority-0 punch-list item 2.
Ch. II — differentiable manifolds
Smooth manifold, atlas, charts	03.02.01-smooth-manifold.md	✓	Single Codex manifold unit. Shared with KN-I §I; nothing to add.
Submersion / immersion / embedding	—	✗	Gap. On the Milnor MMT punch-list.
Tangent / cotangent bundle	—	✗	Gap. Shared with KN-I priority-0 punch-list item 1 (03.02.02).
Regular value theorem	—	✗	Gap. Shared with the Milnor audit.
Partitions of unity	—	✗	Gap. Foundational; nothing in the Codex.
Whitney embedding (weak, $M^n\hookrightarrow {\mathbb{R}}^{2n+1}$)	—	✗	Gap. Sternberg gives the easy version; strong Whitney $M^n\hookrightarrow {\mathbb{R}}^{2n}$ is a Milnor topic. Cite Whitney 1936 originator.
Ch. III — vector fields, Lie groups (early)
Lie bracket $[X,Y]$, Lie derivative	—	✗	Gap. Shared with KN-I priority-0 punch-list item 2 (03.02.03).
Frobenius theorem	—	✗	Gap. Shared with KN-I priority-0 punch-list item 3 (03.02.04).
Lie group, Lie algebra	03.03.01-lie-group.md, 03.04.01-lie-algebra.md	✓	Shipped (algebra unit lives in 04-differential-forms/ for historical reasons; flagged elsewhere).
Group action (free, transitive, proper)	03.03.02-group-action.md	✓	Shipped.
One-parameter subgroup, exponential map	—	✗	Gap. Shared with KN-I priority-0 punch-list item 4 (03.03.04).
Maurer-Cartan form on $G$	—	✗	Gap. Shared with KN-I. LDG's Ch III treatment is more leisurely than KN-I §I.4.
Cartan's closed-subgroup theorem	—	✗	Gap. Distinctive to LDG / Warner. Cite Cartan 1930.
Lie's third theorem (statement; simply-connected case)	—	✗	Gap. Cited from Lie-algebra units without anchor.
Ch. IV — differential forms, exterior derivative, Stokes
Exterior algebra ${\Lambda }^k{T}^{\ast }M$	△	△	Implicit in 03.04.02-differential-forms.md; no dedicated multilinear-algebra unit.
Differential form $\alpha \in {\Omega }^k(M)$	03.04.02-differential-forms.md	✓	Shipped.
Exterior derivative $\mathrm{d}$, axiomatic uniqueness	03.04.04-exterior-derivative.md	✓	Shipped. Master tier covers the axiomatic uniqueness proof; LDG is the canonical source.
Pullback $f^{\ast }\alpha$	△	△	Covered in passing inside 03.04.02 / 03.04.04; no dedicated unit but adequate.
Integration ${\int }_M\alpha$	03.04.03-integration-on-manifolds.md	✓	Shipped.
Stokes' theorem	03.04.05-stokes-theorem.md	✓	Shipped.
de Rham cohomology	03.04.06-de-rham-cohomology.md	✓	Shipped.
de Rham theorem $H_{\mathrm{dR}}^k\cong H^k(M;\mathbb{R})$	△	△	Covered in 03.04.11-cech-de-rham.md via Čech-de Rham; LDG sketches a more direct singular-cohomology bridge.
Ch. V — Riemannian connections and Cartan structural equations
Riemannian metric, Levi-Civita connection	—	✗	Gap. Shared with KN-I priority-1 punch-list item 12 (03.03.X1). Single most-cited missing unit in the Codex.
Moving frame / orthonormal coframe	—	✗	Gap. Distinctive to LDG. The KN-I audit punch-list does not separately call out the moving-frame construction; LDG's Ch. V is the canonical anchor.
Connection 1-forms ${\omega }_j^i$ in a moving frame	—	✗	Gap. Distinctive to LDG.
First structural equation $\mathrm{d}{\theta }^i+{\omega }_j^i\wedge {\theta }^j={\Theta }^i$	—	✗	Gap. Shared with KN-I priority-1 punch-list item 10 (03.05.14, torsion + structural equations) and punch-list item 11 (03.05.15, soldering form). The moving-frame side should be added to those units as the Beginner tier.
Second structural equation $\mathrm{d}{\omega }_j^i+{\omega }_k^i\wedge {\omega }_j^k={\Omega }_j^i$	03.05.09-curvature.md	△	KN-I principal-bundle form is shipped; the moving-frame form is the natural Beginner-tier presentation and is missing.
Algebraic + differential Bianchi identities	△	△	Shared with KN-I priority-1 item 10.
Sectional, Ricci, scalar curvature	—	✗	Gap. Shared with KN-I priority-2 punch-list item 13 (03.02.05).
Geodesic, exponential ${\exp }_p$, normal coordinates	—	✗	Gap. Shared with KN-I and Milnor audits (03.03.X1).
Hopf-Rinow theorem	—	✗	Gap. Shared with KN-I and Milnor.
Ch. VI — $G$-structures and Cartan prolongation (LDG-distinctive)
$G$-structure on $M$ (reduction of $L(M)$ to $G\subseteq \mathrm{GL}(n)$)	△	△	The orthogonal frame bundle 03.05.03-orthogonal-frame-bundle.md is the canonical example; the general $G$-structure unit is absent and is shared with KN-I priority-1 item 8 (03.05.12, reduction of structure group).
Integrability of a $G$-structure	—	✗	Gap (LDG-only at this level). When is a $G$-structure locally equivalent to the flat model?
Cartan's first prolongation $G^{(1)}\subseteq \mathfrak{g}\otimes ({\mathbb{R}}^n{)}^{\ast }$	—	✗	Gap (LDG-only). Algebraic prerequisite for the prolongation algorithm.
Spencer cohomology of $\mathfrak{g}\hookrightarrow \mathfrak{gl}(n)$	—	✗	Gap (LDG-only).
$G$-structures of finite vs infinite type	—	✗	Gap (LDG-only). Riemannian / conformal finite; symplectic / complex infinite.
Guillemin-Sternberg classification of finite-type $G$-structures	—	✗	Gap (LDG-only; Master-tier pointer). Guillemin-Sternberg 1964 originator.
Ch. VII — symplectic structures and classical mechanics
Symplectic manifold $(M,\omega )$	△	△	Covered in 05-classical-mechanics/ and in the Cannas-da-Silva audit (cannas-da-silva-symplectic.md); the Cannas audit is the canonical anchor.
Hamiltonian vector field $X_H$	△	△	Covered in 05-classical-mechanics/; see also the Cannas audit.
Poisson bracket $\{f,g\}$	△	△	Same.
Liouville's theorem (volume preservation)	△	△	Covered in 05-classical-mechanics/; flagged in concept catalog.
Hamilton's principle of stationary action; Euler-Lagrange	△	△	Covered in 05-classical-mechanics/ and in the Arnold-mechanics audit (FT 1.01).
Legendre transform $TM\to T^{\ast }M$	△	△	Same.
Moment map / first traces of geometric mechanics	—	✗	Out of scope for LDG audit. Deferred to Cannas / Guillemin-Sternberg Geometric Asymptotics / FT 1.14.

Aggregate coverage estimate: ~30–35% of LDG has corresponding Codex units. Chapter IV (differential forms + Stokes + de Rham) is the best-covered chapter at ~85% — 03.04.02 through 03.04.07 and 03.04.11 together shadow LDG §§4.1–4.6 closely, and LDG is already implicitly the master anchor for those units. Chapters I–II are ~10% covered at the manifold-prerequisite layer (shared gap with KN-I and Milnor audits). Chapter III (Lie groups, early treatment) is ~30% covered — the Lie-group and Lie-algebra units are shipped, but the load-bearing intermediate units (exponential map, Maurer-Cartan, closed-subgroup theorem) are missing. Chapter V (Riemannian connections) is ~10% covered — the curvature unit ships in principal-bundle form without the moving-frame side. Chapter VI ($G$-structures and prolongation) is ~5% covered — only the orthogonal-frame-bundle specialisation. Chapter VII (symplectic / mechanics) is ~50% covered via the 05-classical-mechanics/ chapter and is scoped out of this audit by the §5 non-goals (the canonical anchor is Cannas-da-Silva + Arnold, not LDG).

Overlap with KN-I (FT 3.18) audit. Substantial — both books cover the Riemannian-connection / curvature material of LDG Ch. V and KN-I Chs. II–IV. Of the 12 KN-I priority-0+1 punch-list items, 9 are shared with this LDG audit at minimum:

KN-I punch-list item	LDG counterpart	Shared verdict
03.02.02 Tangent bundle	LDG Ch. II	Ship once, cite both as anchors.
03.02.03 Lie bracket / Lie derivative	LDG Ch. III	Ship once. LDG provides the Lie-group-flavoured Beginner tier; KN-I provides the principal-bundle-flavoured Master tier.
03.02.04 Frobenius	LDG Ch. III	Ship once. LDG is the canonical anchor (more leisurely than KN-I).
03.03.04 Exponential / Maurer-Cartan	LDG Ch. III	Ship once. LDG is the canonical anchor.
03.05.00 General fibre bundle	LDG Ch. VI (in $G$-structure form)	Ship once. LDG's Ch. VI is the natural Master-tier deepening.
03.05.12 Reduction of structure group	LDG Ch. VI ($G$-structure)	Ship once. Add LDG Ch. VI as the master anchor; KN-I §II.7 as secondary.
03.05.14 Torsion + structural equations	LDG Ch. V	Ship once with two-anchor framing. LDG gives the moving-frame Beginner tier; KN-I §III gives the principal-bundle Master tier.
03.05.15 Soldering form / linear connection	LDG Ch. V	Ship once with two-anchor framing. Same as above.
03.03.X1 Levi-Civita / exp / Hopf-Rinow	LDG Ch. V	Ship once. Also shared with Milnor MMT audit.

Net effect: the LDG audit closes a small number of LDG-distinctive gaps (Ch. VI prolongation algebra, Spencer cohomology, finite-type classification, Cartan closed-subgroup theorem) but mostly adds the moving-frame Beginner tier to units already on the KN-I punch-list. The audit is therefore moderate-sized as a standalone, but with heavy coordination cost against the KN-I audit.

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

Coordination note. Every item below is flagged as either [LDG-only] (no KN-I counterpart, ship from this audit) or [shared with KN-I item N] (ship the unit once and cite both anchors). Shared items are not duplicated production work; they appear here to make the dependency explicit and to record the LDG framing that should be added to the unit (typically as the Beginner tier or as a §Notation moving-frame paragraph).

Priority 0 — manifold-layer and Lie-group-layer prerequisites (all shared with KN-I priority-0):

1. 03.02.02 Tangent bundle as a smooth vector bundle. [shared with KN-I item 1] — KN-I §I.2 + LDG §II as joint master anchors. LDG's Beginner-tier framing (derivation at a point, germ of functions) is the more accessible entry.
2. 03.02.03 Vector fields, Lie bracket, Lie derivative. [shared with KN-I item 2] — KN-I §I.3 + LDG §III.1 joint master anchors. LDG also covers the flow of a vector field in this chapter; recommend folding the flow content here rather than a separate unit.
3. 03.02.04 Frobenius theorem. [shared with KN-I item 3] — LDG §III is the canonical anchor; KN-I §I.2 (Theorem 2.1) is secondary. The pedagogically distinctive Beginner-tier exposition (flow-of-vector-fields + Lie-bracket criterion) follows LDG.
4. 03.03.04 Exponential map $\exp :\mathfrak{g}\to G$ and the Maurer-Cartan form. [shared with KN-I item 4] — LDG §III is the canonical anchor; KN-I §I.4 is secondary; Warner Ch. 3 tertiary.
5. 03.05.00 General fibre bundle. [shared with KN-I item 5] — Steenrod §2–§3 as the master anchor; LDG §VI is the natural Master-tier deepening into the $G$-structure framing.

Priority 1 — LDG-distinctive Ch. III + Ch. V additions:

6. 03.03.05 Cartan's closed-subgroup theorem. [LDG-only] Every closed subgroup $H\subseteq G$ of a Lie group is a Lie subgroup (i.e. an embedded submanifold and a Lie group with the subspace topology). LDG §III anchor; Warner §3.21 secondary; originator Cartan 1930. ~1200 words. Intermediate + Master.
7. 03.03.06 Lie's third theorem (statement, simply-connected case). [LDG-only] Every finite-dim Lie algebra $\mathfrak{g}$ over $\mathbb{R}$ is the Lie algebra of a (unique, up to isomorphism) simply-connected Lie group $G$. LDG §III anchor; Lie's original 1893 reference + Cartan 1930 + Serre Lie Algebras and Lie Groups FT 1.03 secondary anchors. ~1200 words. Master-dominant; Intermediate tier states without proof.
8. 03.05.14 Torsion tensor and the two Cartan structural equations (moving-frame Beginner tier). [shared with KN-I item 10] — LDG §V is the canonical anchor for the moving-frame Beginner tier. The KN-I principal-bundle Master tier remains primary. Notation crosswalk between ${\theta }^i$ / ${\omega }_j^i$ / ${\Theta }^i$ / ${\Omega }_j^i$ (LDG) and $\theta$ / $\omega$ / $\Theta$ / $\Omega$ (KN-I) recorded in a §Notation paragraph.
9. 03.05.15 Linear connection via the frame bundle and the soldering form (moving-frame Beginner tier). [shared with KN-I item 11] — Same two-anchor framing as item 8.
10. 03.03.X1 Levi-Civita connection, geodesics, exponential map, Hopf-Rinow. [shared with KN-I item 12 and Milnor MMT punch-list] — KN-I §IV + LDG §V + Milnor MMT §10 joint master anchors. LDG provides the most pedagogical Beginner tier (moving-frame + explicit Christoffel symbols); KN-I provides the Master tier (principal-bundle + holonomy); Milnor provides the Morse-theoretic bridge to the index theorem.

Priority 2 — LDG-distinctive Ch. VI ($G$-structures and prolongation, the chapter that justifies the audit beyond KN-I):

11. 03.05.12 Reduction of structure group; $G$-structure on $M$. [shared with KN-I item 8] — KN-I §II.7 + LDG §VI joint master anchors. LDG's $G$-structure framing as a reduction of $L(M)$ is pedagogically primary; the abstract reduction theorem from KN-I is the Master-tier statement.
12. 03.05.X1 Integrability of a $G$-structure. [LDG-only] Definition: a $G$-structure $P\subseteq L(M)$ is integrable if every $x\in M$ has a chart in which $P$ is the standard flat $G$-structure on ${\mathbb{R}}^n$. Examples: an $\mathrm{O}(n)$-structure (Riemannian metric) is integrable iff the metric is flat (Riemann curvature vanishes); a $\mathrm{GL}(n, \mathbb{C})$-structure (almost-complex structure) is integrable iff the Newlander-Nirenberg torsion vanishes. LDG §VI anchor; Sternberg-Guillemin 1964 Bull. AMS originator paper. ~1500 words. Intermediate + Master.
13. 03.05.X2 First prolongation $G^{(1)}$ and Spencer cohomology. [LDG-only] Define $G^{(1)} = (\mathfrak{g} \otimes (\mathbb{R}^n)^) \cap (\mathbb{R}^n \otimes S^2 (\mathbb{R}^n)^)$ where the intersection is taken inside the Spencer complex. Iteration $G\supseteq G^{(1)}\supseteq G^{(2)}\supseteq \cdots$. Spencer cohomology $H^{p,q}(\mathfrak{g})$ as the obstruction to further prolongation. LDG §VI anchor; Spencer 1962 (Annals of Math. 76) and Sternberg-Guillemin 1964 originators; Bryant et al. Exterior Differential Systems (Springer 1991) secondary anchor. ~2000 words. Master-dominant.
14. 03.05.X3 $G$-structures of finite vs infinite type. [LDG-only] Definition: $G$-structure is of finite type if some $G^{(k)}=0$, infinite type otherwise. Riemannian ($\mathrm{O}(n)$), conformal ($\mathrm{CO}(n)$), and the projective structures are finite-type; symplectic ($\mathrm{Sp}(2n)$) and complex ($\mathrm{GL}(n,\mathbb{C})$) are infinite-type. Consequence: finite-type structures have finitely many local invariants determined by a finite jet of the structure; infinite-type structures have function-degree-of-freedom local data (whence Darboux's theorem for symplectic structures: no local invariants at all once $\dim M=2n$ is fixed). LDG §VI anchor; Kobayashi Transformation Groups in Differential Geometry (Ergebnisse 1972) secondary. ~1500 words. Intermediate + Master.
15. 03.05.X4 Guillemin-Sternberg classification of irreducible finite-type $G$-structures. [LDG-only; pointer + Master-tier only] Statement of the classification (only $\mathrm{O}(n,p,q)$, $\mathrm{CO}(n,p,q)$, $\mathrm{GL}(n,\mathbb{R})$, $\mathrm{SL}(n,\mathbb{R})$, $\mathrm{Sp}(n,\mathbb{R})$ if we bend the definition of finite-type, and a few exceptional types). Guillemin-Sternberg 1964 Bull. AMS anchor. ~1000 words. Master-only pointer unit. The detailed classification proof is deferred to the research literature.

Priority 3 — Ch. I + Ch. II foundational gaps shared with other audits:

16. 03.02.X1 Submersions, immersions, and the regular value theorem. [shared with Milnor MMT punch-list item 1] — Milnor's Topology from the Differentiable Viewpoint + LDG §II + Lee §5 joint anchors. ~1500 words. Three-tier.
17. 03.02.X2 Partitions of unity. [LDG-only at this audit; also on the Lee audit when ready] — LDG §II + Lee §13 anchors. ~1000 words. Intermediate + Master.
18. 03.02.X3 Sard's theorem. [shared with Milnor MMT punch-list] — Sard 1942 originator; Milnor Topology from the Differentiable Viewpoint §3 anchor; LDG §I secondary. ~1500 words. Three-tier.
19. 03.02.X4 Whitney embedding theorem (weak, $M^n\hookrightarrow {\mathbb{R}}^{2n+1}$). [LDG-only at this audit; could be a Milnor punch-list item] — LDG §II + Whitney 1936 originator. ~1500 words. Intermediate + Master. The strong Whitney embedding ($M^n\hookrightarrow {\mathbb{R}}^{2n}$) is deferred to a Milnor Topology from the Differentiable Viewpoint audit.

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

20. Pointer in 03.05.03-orthogonal-frame-bundle.md to the new 03.05.12 and 03.05.X1 as the general $G$-structure context (with $\mathrm{O}(n)$-structure as the canonical example). Single-paragraph weaving edit.
21. Pointer in cannas-da-silva-symplectic.md audit and the 05-classical-mechanics/ chapter to 03.05.X3 for the symplectic is infinite-type explanation of Darboux's theorem. Single-paragraph weaving edit.
22. Pointer in the Sternberg later-book audits (FT 1.14 Curvature in Math and Physics, FT 1.15 Group Theory and Physics) to the Ch. VI prolongation units as the upstream Sternberg-tradition anchor. Recorded so it is not forgotten in those audits.

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§4 Implementation sketch (P3 → P4)

For a full LDG coverage pass, priority-0+1 are the minimum set, but priority-0+1 are almost entirely shared with KN-I; the LDG-only production work begins at priority-1 items 6–7 and priority-2 items 12–15. Realistic production estimate:

• Priority 0: 5 units × ~2.5 hours = ~12 hours. Shared with KN-I audit; do not double-count. Counted toward the LDG audit only for the §Notation moving-frame additions and the §1-prose LDG anchor.
• Priority 1: 5 units × ~3.5 hours = ~17 hours, of which 3 units (Cartan closed-subgroup, Lie's third theorem; moving-frame additions to KN-I items 10/11; the joint Levi-Civita item) are LDG-distinctive net additions = ~10 hours of LDG-only production.
• Priority 2: 5 units × ~4 hours = ~20 hours. All LDG-only. This is the chapter that justifies the Sternberg audit beyond the KN-I audit. Master-tier dominant because the Spencer-cohomology and prolongation algebra is technical.
• Priority 3: 4 units × ~2.5 hours = ~10 hours. Shared with Milnor and Lee audits.
• Priority 4: 3 weaving edits × ~30 min = ~1.5 hours.
• LDG-only total (counting only net new units): ~32 hours of focused production (priority-2 chapter + Cartan closed-subgroup + Lie's third + the moving-frame additions). Fits a focused 4–5 day window.
• Full-coverage total (counting shared units once): ~60 hours.

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

• Élie Cartan, "Sur la structure des groupes de transformations finis et continus," thèse, Paris 1894 — the originator paper for the moving-frame method and the structural equations. Cite in 03.05.14 and 03.05.15.
• 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 development of affine connections via moving frames. Cite in 03.05.14, 03.05.15 alongside the 1894 thesis. Shared with KN-I originator-citation list.
• Élie Cartan, "La théorie des groupes finis et continus et l'analysis situs," Mémorial des sciences mathématiques 42 (1930) — the closed-subgroup theorem; cite in 03.03.05.
• Hassler Whitney, "Differentiable manifolds," Annals of Math. 37 (1936) 645–680 — the weak embedding theorem $M^n \hookrightarrow \mathbb{R}^{2n+1}$; cite in 03.02.X4.
• Charles Ehresmann, "Les connexions infinitésimales dans un espace fibré différentiable," Colloque de topologie de Bruxelles (1950) — formal connection on a principal bundle; shared with KN-I originator list. Cite in 03.05.07 and 03.05.11.
• Donald Spencer, "Deformation of structures on manifolds defined by transitive, continuous pseudogroups," Annals of Math. 76 (1962) 306–445 — Spencer cohomology; cite in 03.05.X2.
• Victor Guillemin, Shlomo Sternberg, "An algebraic model of transitive differential geometry," Bull. AMS 70 (1964) 16–47 — the classification of finite-type irreducible $G$-structures; cite in 03.05.X2, 03.05.X3, 03.05.X4.
• Shlomo Sternberg, Lectures on Differential Geometry, Prentice-Hall 1964 — the canonical consolidation; cite throughout as the in-Codex master anchor for Ch. V (moving-frame Beginner tier), Ch. VI ($G$-structures), and as a joint anchor with KN-I for the Riemannian-connection units.

Notation crosswalk. LDG writes:

• $(e_i)$, $({\theta }^i)$ for moving frame and dual coframe on $M$ (orthonormal when $G=\mathrm{O}(n)$); KN-I uses $(\theta )$ for the soldering 1-form and never names the frame field explicitly.
• ${\omega }_j^i$ for connection 1-forms in the moving frame; KN-I writes $\omega =({\omega }_j^i)\in {\Omega }^1(L(M);\mathfrak{gl}(n))$ as the principal-bundle connection 1-form.
• ${\Theta }^i$ for the torsion 2-forms in the moving frame; ${\Omega }_j^i$ for curvature 2-forms. KN-I uses $\Theta$ and $\Omega$ as the matrix / Lie-algebra-valued forms.
• LDG writes $G^{(k)}$ for the $k$-th prolongation of $\mathfrak{g} \subseteq \mathfrak{gl}(n, \mathbb{R})$; nothing analogous in KN-I.

The Codex notation decision (per docs/specs/UNIT_SPEC.md §11) should: adopt KN-I's $\omega$, $\Omega$, $\Theta$, $\theta$ verbatim as the Master-tier presentation; introduce LDG's ${\omega }_j^i$, ${\Omega }_j^i$, ${\Theta }^i$, ${\theta }^i$, $(e_i)$ as the Beginner-tier presentation in the same units (03.05.14, 03.05.15, 03.03.X1); record the correspondence in a §Notation paragraph stating that the principal- bundle form is the pullback of the moving-frame form along a local section $M\to L(M)$. For the prolongation units (03.05.X1–03.05.X4) adopt LDG's $G^{(k)}$ notation verbatim — there is no KN-I counterpart.

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§5 What this plan does NOT cover

• A line-number-level inventory of every named theorem in LDG (full P1 audit). Deferred until a local PDF is on disk in reference/fasttrack-texts/01-fundamentals/Sternberg-LecturesOnDifferentialGeometry.pdf (the natural location, since FT 1.10 is a §1 fundamentals entry). This stub works from canonical TOC knowledge + Codex internal evidence and is therefore REDUCED.
• Sternberg, Curvature in Mathematics and Physics — Fast Track 1.14 (Dover 2012). Deferred to a separate audit. Covers GR + gauge theory at the geometric-mechanics level; the Ch. VI prolongation apparatus of LDG is its upstream anchor.
• Sternberg, Group Theory and Physics — Fast Track 1.15 (CUP 1994). Deferred to a separate audit. Covers representation theory applied to quantum mechanics; partial overlap with Fulton-Harris and Woit audits.
• Symplectic + Hamiltonian mechanics (LDG Ch. VII). The canonical Codex anchor for the symplectic side is Cannas-da-Silva (cannas-da-silva-symplectic.md); for the Hamiltonian-mechanics side it is Arnold Mathematical Methods of Classical Mechanics (FT 1.01). LDG Ch. VII is a preview; substantive coverage belongs to those audits.
• Geometric mechanics in the Marsden-Weinstein / moment-map / reduction sense. Deferred to Cannas-da-Silva + the future Marsden-Ratiu Introduction to Mechanics and Symmetry audit if catalogued.
• Exterior differential systems / EDS (Bryant-Chern-Gardner- Goldschmidt-Griffiths). LDG Ch. VI is the conceptual ancestor of EDS, but the modern EDS treatment is its own subject; deferred to a potential Bryant et al. audit, not on the current Fast Track list.
• Riemannian comparison geometry, Ricci flow, geometric analysis. Far downstream of LDG. Shared with the KN-I and Jost audits.
• Pseudo-Riemannian / Lorentzian geometry. LDG is positive-definite throughout. Pseudo-Riemannian belongs to the Semi-Riemannian Geometry Sternberg-PDF (FT-adjacent, locally available at reference/fasttrack-texts/01-fundamentals/Sternberg-SemiRiemannianGeometry.pdf) and to O'Neill's textbook; deferred.

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§6 Acceptance criteria for FT equivalence (LDG)

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

• ≥95% of LDG's named theorems in Chs. I–VI map to Codex units. Ch. VII is scoped out (covered by Cannas + Arnold audits). Current ~30%; after priority-0+1 rises to ~75%; after priority-0+1+2 to ~93%; full ≥95% requires priority-3.
• ≥90% of LDG's worked examples in Chs. I–VI have either a direct unit or are referenced from a unit covering them. The canonical LDG worked examples are: the orthonormal-coframe computation on the round sphere $S^n\subseteq {\mathbb{R}}^{n+1}$ (Ch. V); the Maurer-Cartan form on $\mathrm{GL}(n,\mathbb{R})$ (Ch. III); the flat-model integrability check for each of the standard $G$-structures (Ch. VI).
• The moving-frame Beginner-tier additions are present in 03.05.09, 03.05.14, 03.05.15, 03.03.X1 per §3 items 8–10.
• The Ch. VI prolongation chain (03.05.X1–03.05.X4) ships as a contiguous four-unit cluster, weaving back to 03.05.03-orthogonal- frame-bundle.md and to the Cannas-da-Silva audit (per §3 priority-4 weaving items).
• Notation decisions (moving-frame vs principal-bundle conventions) are recorded in 03.05.14, 03.05.15, 03.05.X1 per §4.
• Originator-prose citations of Cartan 1894 / 1923–25 / 1930, Whitney 1936, Ehresmann 1950, Spencer 1962, Guillemin-Sternberg 1964, and Sternberg 1964 are present in the relevant units.
• Shared-with-KN-I items ship exactly once with joint anchor citations; the KN-I and LDG audits do not double-count production work.

The 5 priority-0 + 5 priority-1 units close the bulk of the shared manifold + Lie-group + Riemannian-connection gap (most of which is already on the KN-I punch-list). Priority-2 closes the LDG-only distinctive content ($G$-structures + prolongation + Spencer cohomology + Guillemin-Sternberg classification) — this is what makes the audit worth doing as a separate pass rather than folding into KN-I. Priority-3 closes the foundational Ch. I + Ch. II gaps shared with the Milnor audit.

What survives as Sternberg-only content after KN-I and Jost have shipped? The Ch. VI prolongation chain (items 12–15: integrability, Spencer cohomology, finite vs infinite type, Guillemin-Sternberg classification), the Cartan closed-subgroup theorem (item 6) and Lie's third theorem (item 7) at the level developed in LDG Ch. III, and the moving-frame Beginner-tier presentation of the Cartan structural equations in 03.05.14–03.05.15 and 03.03.X1. Everything else either ships once on the KN-I punch-list or is scoped out to the Cannas / Arnold / Milnor audits. The Ch. VI prolongation cluster is the single strongest argument for a separate Sternberg audit; without it, LDG would reduce to "the moving-frame Beginner tier of the KN-I units" and the audit could fold into KN-I.

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§7 Sourcing

• Not free. LDG is under active Chelsea / AMS copyright (Chelsea reprint 1983, AMS-Chelsea Series, ISBN 0-8284-0316-3). The author's Harvard page (people.math.harvard.edu/~shlomo/) hosts six other Sternberg books — Real Variables, Advanced Calculus (the preferred LDG prerequisite), Dynamical Systems, Lie Algebras, Semi-Riemannian Geometry, Semi-Classical Analysis — but explicitly does not host Lectures on Differential Geometry. WebFetch on the Harvard page (2026-05-17) confirmed the omission. Print copies (AMS-Chelsea reprint) are readily available second-hand in the $40–80 range.
• Local copy. Not present in reference/fasttrack-texts/01-fundamentals/ at the time of this audit. The five Sternberg PDFs already on disk are the later books (Advanced Calculus, Dynamical Systems, Lie Algebras, Semi-Riemannian, Semi-Classical) — none is LDG. A scanned copy should be added to reference/fasttrack-texts/01-fundamentals/Sternberg-LecturesOnDifferentialGeometry.pdf before the production pass on the priority-2 punch-list units, so line-number citations to LDG Ch. VI can be resolved.
• 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 the LDG Ch. V principal-bundle material in moving-frame and principal- bundle parallel form. Preferred secondary anchor for the shared KN-I priority-1 units.
  • M. Spivak, A Comprehensive Introduction to Differential Geometry, Vols. I–II, Publish or Perish (3rd ed. 1999). Vol. I Chs. 1–6 parallels LDG Chs. I–II at a slower pace with extensive figures; Vol. II Chs. 7–8 parallels LDG Ch. V (Cartan structural equations). The Spivak treatment of $G$-structures (Vol. II Ch. 9 "what connections are good for") is the closest peer treatment of LDG Ch. VI in print.
  • M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992. The standard graduate Riemannian-geometry text; covers LDG Ch. V at a more leisurely pace, with the moving-frame computation on $S^n$ as a worked example. Peer anchor for 03.03.X1 Beginner tier.
  • W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press (2nd revised ed. 2003). Canonical American 1970s textbook that follows the LDG ordering (Lie groups before differential forms). Peer anchor for the shared priority-0 manifold and Lie-group units.
  • F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics 94, Springer 1983. The other canonical American 1970s textbook in the LDG tradition; Ch. 3 is the cleanest exposition of the Cartan closed-subgroup theorem and Lie's third theorem at the level required for the priority-1 items 6–7.
  • R. Bryant, S.-S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths, Exterior Differential Systems, MSRI Publications 18, Springer 1991. The modern EDS treatment; Ch. III + Appendix A cover Spencer cohomology and the prolongation algorithm at full research depth. Cite as the master Master-tier anchor for 03.05.X2.
  • S. Kobayashi, Transformation Groups in Differential Geometry, Ergebnisse der Mathematik 70, Springer 1972. Covers $G$-structures and the prolongation algorithm at a level between LDG Ch. VI and Bryant et al. EDS. Cite in 03.05.X1–03.05.X4 as a secondary anchor.
  • Sternberg's own later works (FT 1.04 Advanced Calculus — locally available; FT 1.14 Curvature in Math and Physics — to be sourced; FT 1.15 Group Theory and Physics — to be sourced) are the downstream Sternberg-tradition continuations.
• Originator-paper archive locations:
  • Cartan 1894 thèse — open access via NUMDAM / Œuvres complètes.
  • Cartan 1923–25 Annales sci. ÉNS — open access via NUMDAM.
  • Cartan 1930 Mémorial sci. math. — open access via NUMDAM.
  • Whitney 1936 Annals of Math. — JSTOR.
  • Ehresmann 1950 Colloque de topologie de Bruxelles — collected works; print only.
  • Spencer 1962 Annals of Math. — JSTOR.
  • Guillemin-Sternberg 1964 Bull. AMS — AMS open archive.
  • Sternberg 1964 LDG itself — AMS-Chelsea reprint; copyright; print only (and as the audit target, the priority sourcing item).