Jost — Riemannian Geometry and Geometric Analysis (Fast Track 3.28) — Audit + Gap Plan

Book: Jürgen Jost, Riemannian Geometry and Geometric Analysis (Springer Universitext, 7th edition 2017, xiv + 697 pp.; ISBN 978-3-319-61859-3). The standing successor to do Carmo / Petersen / Lee on the analytic side: a Riemannian-geometry text that pushes through to harmonic maps, Yamabe, and parabolic methods rather than stopping at comparison theorems.

Fast Track entry: 3.28 (third-decile modern geometry, marked BUY in docs/catalogs/FASTTRACK_BOOKLIST.md line 115). Sits between the classical-Riemannian texts (do Carmo, Petersen, Lee) and the geometric-analysis monographs (Aubin, Schoen-Yau, Hamilton).

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2

P3-lite of the orchestration protocol) producing a concrete punch-list. Output is the canonical closing pass on the basic-Riemannian-geometry gap in the Codex — a gap previously flagged without remedy by three earlier audits (Milnor MMT, Helgason DGLGSS, Kobayashi-Nomizu I).

Sourcing status: Reduced. No local PDF in reference/textbooks-extra/ or reference/fasttrack-texts/; no free author copy (Springer commercial title). Plan is built from Jost's canonical chapter structure (well-documented in the Springer ToC and in the citation record of Aubin, Schoen-Yau, and Hebey) and from the existing Codex audit chain. A line-number-precise pass requires the book on hand and is deferred.

§1 What Jost is for

Jost is the canonical modern bridge from Riemannian geometry to geometric analysis. Other Riemannian texts at the same level — do Carmo Riemannian Geometry (Birkhäuser 1992), Petersen Riemannian Geometry (GTM 171, Springer 3rd ed. 2016), Lee Introduction to Riemannian Manifolds (GTM 176, Springer 2nd ed. 2018) — all stop at the comparison theorems (Bonnet-Myers, Cartan-Hadamard, Toponogov, Rauch) and a survey of curvature topology. Cheeger-Ebin Comparison Theorems in Riemannian Geometry (AMS Chelsea reprint 2008) goes deeper into comparison geometry but also stops at the finite-dimensional toolkit. Jost is the text that keeps going into the infinite-dimensional / PDE side.

The book's distinctive structure, in roughly the order Jost develops it:

Foundations (Chs. 1–3). Riemannian manifolds, the Levi-Civita connection, geodesics, the exponential map, normal coordinates, completeness (Hopf-Rinow theorem, Hopf-Rinow 1931). Standard first-course content, but with Jost's analytic-PDE notation conventions ( $g_{ij}$ and $Γ_{ij}^{k}$ throughout, weak emphasis on the principal-bundle reformulation that Kobayashi-Nomizu prefer).
Curvature (Chs. 4–5). Riemannian curvature tensor $R$ , sectional, Ricci, and scalar curvature. Jacobi fields, conjugate points, the second variation of energy. The comparison theorems: Bonnet-Myers (Myers 1941), Cartan-Hadamard (Cartan 1925–26 / Hadamard 1898), Toponogov (Toponogov 1959). Cf. Petersen Chs. 6–7 and Cheeger-Ebin Chs. 1–4 for parallel treatments [ref: Petersen, Riemannian Geometry GTM 171, 3rd ed. 2016, §§6–7; Cheeger-Ebin, Comparison Theorems in Riemannian Geometry AMS Chelsea 2008, Chs. 1–4].
Symmetric spaces and Lie groups (Ch. 6). A short symmetric-space chapter giving the Riemannian-geometric face of what Helgason DGLGSS does Lie-algebraically. Significantly thinner than Helgason but sufficient to motivate the comparison-geometry models.
Spin structures and the Dirac operator (Ch. 7). Concise introduction; the Codex has this material in 03.09-spin-geometry/ already (Lawson-Michelsohn anchor).
Morse theory and Floer homology (Ch. 8). A working-mathematician sketch of Morse-on-loop-spaces, Bott periodicity via geodesics, pointer to Floer. Parallel to Milnor MMT (covered in Cycle 3 audit) and Schwarz / Audin-Damian (covered in 05.08-floer-symplectic/).
Harmonic maps (Chs. 9–10). The book's first analytic centerpiece. The harmonic-map energy $E (u) = \frac{1}{2} \int_{M} ∣ d u ∣^{2}$ for $u : M \to N$ , the harmonic-map equation $τ (u) = 0$ , the Eells-Sampson theorem (Eells-Sampson 1964) on existence of harmonic maps into nonpositively curved targets via the harmonic- map heat flow $\partial_{t} u = τ (u)$ . The full analytic proof: short-time existence, Bochner identity, energy decay, sub-convergence. This is the chapter no other Riemannian textbook covers in full.
Sobolev spaces on manifolds and elliptic regularity (Ch. 11). $W^{k, p} (M, g)$ defined intrinsically, Sobolev embedding theorems on compact manifolds, Schauder estimates, weak solutions, elliptic regularity for $Δ_{g}$ . Cf. Aubin, Some Nonlinear Problems in Riemannian Geometry (Springer 1998) Ch. 2 and Hebey, Sobolev Spaces on Riemannian Manifolds (LNM 1635, Springer 1996) for the parallel treatments [ref: Aubin 1998 Ch. 2; Hebey 1996 Chs. 1–4].
The Yamabe problem (Ch. 12). The analytic capstone. The Yamabe conjecture (Yamabe 1960) — every compact Riemannian manifold of dimension $\geq 3$ admits a conformally equivalent metric of constant scalar curvature — proved in three rounds: Trudinger 1968 (low-dim), Aubin 1976 (dim $\geq 6$ , non-locally- conformally-flat), Schoen 1984 (remaining cases via the positive mass theorem of Schoen-Yau 1979). Jost gives the full variational / concentration-compactness proof; this is the second-canonical exposition (Lee-Parker The Yamabe Problem Bull. AMS 17 (1987) 37–91 is the first).

Where Jost differs from do Carmo, Petersen, and Lee: those texts cover items 1–2 (and partially 3) and stop; Jost's items 6–8 are the Springer-Universitext analytic capstone that closes the loop with the PDE-and-calculus-of-variations side of modern geometry. Jost is also the text most often cited as the prerequisite anchor for geometric-flow texts (Hamilton-Perelman / Topping Lectures on the Ricci Flow / Chow-Knopf): students arriving at Ricci flow are expected to know Jost Chs. 9 and 11 before opening Hamilton's papers.

Jost is not a first introduction to manifolds. He assumes smooth manifolds, vector bundles, and basic functional analysis (Banach / Hilbert spaces, weak convergence). The Codex prerequisite layer for a Jost-anchored unit batch is therefore: 03.02-manifolds/, 03.05-bundles/, and 02.11-functional-analysis/.

§2 Coverage table (Codex vs Jost)

Cross-referenced against the current 313-unit corpus. ✓ = covered, △ = partial / different framing, ✗ = not covered.

Jost topic	Codex unit(s)	Status	Note
Ch. 1 — Riemannian manifold, metric tensor, isometry	—	✗	Gap. Metric tensor as a positive-definite section of $S^{2} T^{*} M$ ; no dedicated unit. The pieces exist in `03.05.03-orthogonal-frame-bundle.md` (frame side) but not the metric-tensor side.
Ch. 2 — Levi-Civita connection	—	✗	Gap (critical). Already on the Milnor MMT punch-list (item 6, `03.03.X1`) and the KN-I punch-list (item 12, `03.03.X1`). The single most-cited missing unit in the Codex per the KN-I audit.
Ch. 2 — Geodesics, geodesic equation $\overset{γ}{¨}^{k} + Γ_{ij}^{k} \overset{γ}{˙}^{i} \overset{γ}{˙}^{j} = 0$	△	△	Geodesic flow appears in `05.02.06-geodesic-flow-hamiltonian.md` from the Hamiltonian-symplectic side; the Riemannian framing is absent. Gap.
Ch. 2 — Exponential map $exp_{p}$ at a point	—	✗	Gap. Cited downstream without anchor (Cartan-Hadamard, Bonnet-Myers, KN-I audit).
Ch. 2 — Normal coordinates	—	✗	Gap.
Ch. 3 — Hopf-Rinow theorem	—	✗	Gap (critical). Already on Milnor MMT, Helgason DGLGSS, and KN-I punch-lists. The canonical example of an audit-chain-closing gap.
Ch. 3 — Cut locus and injectivity radius	—	✗	Gap.
Ch. 4 — Riemannian curvature tensor $R (X, Y) Z$	△	△	Covered as a bundle-curvature object in `03.05.09-curvature.md`; the Riemannian-tensor specialisation (the four symmetries $R_{ij k l}$ , the algebraic Bianchi identity) is partial.
Ch. 4 — Sectional curvature	—	✗	Gap. On the KN-I punch-list (item 13, `03.02.05`).
Ch. 4 — Ricci tensor and scalar curvature	—	✗	Gap. On the KN-I punch-list (item 13, `03.02.05`).
Ch. 5 — Jacobi fields, conjugate points	—	✗	Gap. On the Milnor MMT punch-list (item 5, `03.02.05`).
Ch. 5 — Second variation of energy	—	✗	Gap. On the Milnor MMT punch-list (item 4, `03.12.X1`).
Ch. 5 — Bonnet-Myers theorem	—	✗	Gap. On the Milnor MMT punch-list (item 8, `03.02.06`).
Ch. 5 — Cartan-Hadamard theorem	—	✗	Gap. On the Milnor MMT punch-list (item 8, `03.02.06`).
Ch. 5 — Toponogov triangle comparison	—	✗	Gap. Not on any prior punch-list; new to this audit.
Ch. 5 — Rauch comparison theorem	—	✗	Gap. Not on any prior punch-list; new to this audit.
Ch. 5 — Synge's theorem; first cohomology of compact $sec > 0$	—	✗	Gap (low). Master-tier only.
Ch. 6 — Riemannian symmetric spaces	—	✗	Gap. Already on Helgason punch-list (`07.04.07`). Jost gives the Riemannian-geometric face; Helgason gives the Lie-algebraic. Cross-link the two units in Pass-W.
Ch. 7 — Spin structures, Dirac operator	`03.09.04`, `03.09.08`, `03.09.14`	✓	Covered via Lawson-Michelsohn. No gap.
Ch. 8 — Morse theory on loop spaces, Bott periodicity sketch	`03.08.07`, partial in `12-homotopy/`	△	Covered via the Milnor MMT audit's punch-list. Jost adds nothing new beyond the MMT batch.
Ch. 9 — Harmonic map energy $E (u)$ , harmonic-map equation $τ (u) = 0$	—	✗	Gap (high — Jost's signature topic).
Ch. 9 — Bochner identity for harmonic maps	—	✗	Gap.
Ch. 10 — Harmonic-map heat flow $\partial_{t} u = τ (u)$ ; Eells-Sampson theorem	—	✗	Gap (high — Eells-Sampson 1964 is a top-tier originator citation, no Codex anchor at present).
Ch. 11 — Sobolev spaces $W^{k, p} (M, g)$ on manifolds	`02.11.04-banach-spaces.md` (flat), — (intrinsic)	△	The flat-space functional-analysis layer exists; the intrinsic Riemannian formulation does not. Gap.
Ch. 11 — Sobolev embedding on compact manifolds	—	✗	Gap.
Ch. 11 — Elliptic regularity for $Δ_{g}$	△ (in `06.04.05-dbar-hilbert-pde.md`)	△	The Riemann-surface case is shipped; the higher-dim Riemannian case is not.
Ch. 12 — Conformal class, conformal Laplacian, Yamabe invariant	—	✗	Gap.
Ch. 12 — Yamabe problem (Yamabe 1960 / Trudinger 1968 / Aubin 1976 / Schoen 1984)	—	✗	Gap (high — capstone theorem with a four-decade originator chain).
Ch. 12 — Positive mass theorem (used in Schoen's proof)	`03.09.17-witten-positive-mass.md`	△	Shipped via Witten's spinor proof; Schoen-Yau 1979 (the original proof, used in the Yamabe argument) is the missing originator anchor.

Aggregate coverage estimate. Of Jost's 12 main chapters, the Codex covers Ch. 7 in full, Ch. 8 partially (via prior MMT audit), and items in Ch. 6 already on Helgason's open punch-list. Chapters 1–5 (basic Riemannian geometry) are uncovered. Chapters 9–12 (the geometric-analysis capstone — Jost's signature material) are ~0% covered. This audit's punch-list is the canonical closing pass for Chs. 1–5 (closing the audit-chain Riemannian gap) and the canonical opening pass for Chs. 9–12 (new geometric-analysis units).

Audit-chain status. The Cycle 3 audit (Milnor MMT) flagged Levi-Civita / exp / Hopf-Rinow as item 03.03.X1; the Cycle 4 audit (KN-I) re-flagged it as 03.03.X1 and added sectional / Ricci / scalar (03.02.05); the Cycle 5 audit (Helgason DGLGSS) re-flagged Hopf-Rinow as 03.02.0X. All three audits deferred the basic- Riemannian batch to "the Riemannian-geometry text audit," which is this one. The Jost audit is therefore the convergence point for three prior audit chains and is load-bearing for any future work in gauge theory, geometric flows, Ricci flow / Perelman, harmonic analysis on manifolds, or general relativity.

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

Priority 0 — manifold-layer prerequisites (shared with KN-I audit):

Already on the KN-I punch-list (kobayashi-nomizu-foundations-vol1.md items 1–5). Reproduced here for reading-order completeness; do not duplicate the work.

03.02.02 Tangent bundle as smooth vector bundle.
03.02.03 Vector fields, Lie bracket, Lie derivative.
03.05.00 General fibre bundle.
(No 03.03.04 Lie-group exp needed for Jost specifically; it's a Helgason / KN-I prereq, not a Jost prereq.)

Priority 1 — basic Riemannian geometry (Chs. 1–5; closes the audit-chain gap shared with Milnor / Helgason / KN-I):

03.02.0R1 Riemannian metric tensor and isometry. Definition of $g$ as a smooth section of $S^{2} T^{*} M$ with $g_{p}$ positive definite, pullback under diffeomorphism, isometry. Standard examples (round sphere, hyperbolic space, flat torus). Jost §1.4 anchor; do Carmo Ch. 1 secondary; Lee Ch. 2 secondary. ~1500 words. Three-tier.
03.03.X1 Levi-Civita connection, geodesics, exponential map, Hopf-Rinow. The single most-cited missing unit in the Codex. Shared with Milnor (item 6) and KN-I (item 12) punch-lists; this audit confirms the Jost framing as the canonical anchor (Jost §§2.1–3.1). Includes: fundamental theorem of Riemannian geometry (existence + uniqueness of Levi-Civita), Christoffel symbols $\Gamma^k_{ij} = \frac12 g^{kl}(\partial_i g_{jl} + \partial_j g_{il}
- \partial_l g_{ij}) $, g eo d es i ce q u a t i o n,$ \exp_p$ as a local diffeomorphism near $0 \in T_{p} M$ , Gauss lemma, normal coordinates, Hopf-Rinow (geodesic completeness ⇔ metric completeness ⇔ closed- and-bounded compactness; existence of minimising geodesics). Jost §§2.1–3.1 anchor; do Carmo Chs. 2–7 secondary; Lee Chs. 4–6 secondary; Petersen Chs. 5–6 tertiary. ~3000 words; the longest unit in the punch-list. Three-tier. Originator-prose required (Levi-Civita 1917, Hopf-Rinow 1931).
03.02.0R2 Sectional, Ricci, and scalar curvature. Already on KN-I punch-list as 03.02.05. The three standard curvature contractions, the $K(\sigma) = \langle R(X,Y)Y, X\rangle / |X \wedge Y|^2$ formula, Ricci as a trace, scalar as a double trace. Jost §4 anchor; KN-I §IV.1 secondary. ~1500 words. Three-tier.
03.02.0R3 Jacobi fields and conjugate points. Already on Milnor punch-list as 03.02.05. Jacobi equation as the linearised geodesic equation, conjugate points as zeros of Jacobi fields, relation to $d exp_{p}$ . Jost §5.2 anchor; Milnor MMT §§13–14 secondary; do Carmo Ch. 5 tertiary. ~2000 words. Three-tier.
03.02.0R4 Comparison theorems: Bonnet-Myers and Cartan-Hadamard. Already on Milnor punch-list as 03.02.06. Both as second-variation consequences. Jost §5.3 anchor; Milnor §§19, 22 secondary. ~1500 words. Three-tier. Originator-prose (Myers 1941; Cartan 1925–26 / Hadamard 1898).
03.02.0R5 Rauch and Toponogov comparison theorems. New to this audit (not on prior punch-lists). Rauch: pointwise comparison of Jacobi fields under sectional-curvature bound; Toponogov: global triangle comparison. Jost §5.4 anchor; Cheeger-Ebin Chs. 1–2 secondary. ~1500 words. Three-tier; Master tier carries the proof sketch, Beginner gives only the statements and the "fatter triangles in negative curvature" picture. Originator-prose (Rauch 1951; Toponogov 1959).

Priority 2 — harmonic maps and geometric analysis core (Chs. 9–11; Jost's signature material, no prior Codex anchor):

03.02.0R6 Harmonic map energy, tension field, harmonic-map equation. $E (u) = \frac{1}{2} \int_{M} ∣ d u ∣_{g}^{2} d V_{g}$ for $u : (M, g) \to (N, h)$ , the tension field $\tau(u) = \mathrm{tr}_g \nabla \mathrm{d}u $, t h e E u l er - L a g r an g ee q u a t i o n$ \tau(u) = 0$. Examples: harmonic functions ( $N = R$ ), geodesics ( $M = [0, 1]$ ), holomorphic maps (Kähler case). Jost §9.1 anchor; Eells-Sampson 1964 originator. ~2000 words. Three-tier.
03.02.0R7 Bochner identity for harmonic maps. $\frac12 \Delta |\mathrm{d}u|^2 = |\nabla \mathrm{d}u|^2 + \langle \mathrm{Ric}^M \mathrm{d}u, \mathrm{d}u\rangle - \langle R^N(\mathrm{d}u, \mathrm{d}u) \mathrm{d}u, \mathrm{d}u\rangle$. The curvature-control identity that drives all harmonic-map existence theorems. Jost §9.2 anchor; Eells-Sampson 1964 primary. ~1500 words. Master-dominant, Intermediate carries the statement.
03.02.0R8 Harmonic-map heat flow and the Eells-Sampson theorem. $\partial_{t} u = τ (u)$ ; for $u_{0} : M \to N$ with $M$ compact and $N$ compact of nonpositive sectional curvature, the flow exists for all time and sub-converges to a harmonic map homotopic to $u_{0}$ . The full analytic proof: short-time existence (linear parabolic), Bochner-controlled energy decay, sub-convergence via Arzelà-Ascoli + Schauder. Jost §§10.1–10.3 anchor; Eells-Sampson 1964 originator. ~3000 words. Three-tier; Beginner tier gives the statement and the picture; Intermediate gives the energy-decay argument; Master gives the full proof. Originator-prose required (Eells-Sampson 1964).
03.02.0R9 Sobolev spaces $W^{k, p} (M, g)$ on a Riemannian manifold. Intrinsic definition (covariant derivatives in $L^{p}$ ), equivalence with chart-by-chart definition for compact $M$ , completeness, density of smooth functions. Jost §11.1 anchor; Aubin 1998 §2.1 secondary; Hebey 1996 Ch. 2 tertiary. ~1500 words. Three-tier.
03.02.0R10 Sobolev embedding and Rellich-Kondrachov on compact manifolds. $W^{k, p} (M) ↪ L^{q} (M)$ for the appropriate exponent, compact embedding when the inequality is strict. Jost §11.2 anchor; Aubin §2.5 secondary. ~1500 words. Three-tier; the Beginner tier states the embedding theorem diagram, Master gives the proof via partition of unity.
03.02.0R11 Elliptic regularity for $Δ_{g}$ on a compact manifold. Schauder estimates, $L^{p}$ estimates, weak solutions are smooth. Jost §11.3 anchor. ~1500 words. Three-tier; Master-dominant.

Priority 3 — Yamabe (Ch. 12; capstone theorem):

03.02.0R12 Conformal class, conformal Laplacian, Yamabe invariant. $g \mapsto e^{2 f} g$ , the conformal Laplacian $L_{g} = Δ_{g} - c_{n} Scal_{g}$ ( $c_{n} = (n - 2) / (4 (n - 1))$ ), the Yamabe invariant $Y(M, [g]) = \inf_{\tilde g \in [g]} \mathrm{Vol}(\tilde g)^{-(n-2)/n} \int \mathrm{Scal}{\tilde g} \mathrm{d}V{\tilde g}$. Jost §12.1 anchor. ~1500 words. Three-tier.
03.02.0R13 The Yamabe problem. Statement, the variational setup, the three-round proof outline (Trudinger 1968 / Aubin 1976 / Schoen 1984), the role of the positive mass theorem. Jost §§12.2–12.3 anchor; Lee-Parker Bull. AMS 17 (1987) 37–91 secondary. ~3500 words; the longest unit in the punch-list after 03.03.X1. Three-tier; Master tier carries the full Aubin-test-function calculation and the Schoen positive-mass invocation. Originator-prose required (Yamabe 1960, Trudinger 1968, Aubin 1976, Schoen 1984).

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

03.02.0R14 Synge's theorem. Compact, even-dimensional, $\sec

0$, oriented ⇒ simply connected. Master-only, ~800 words. Jost §5.5 anchor.
Pointer in 03.09.17-witten-positive-mass.md to Schoen-Yau 1979 (the original positive mass theorem, not Witten's spinor proof) and to the new 03.02.0R13 (Yamabe). Single-paragraph weaving edit.
Pointer in 03.05.09-curvature.md to 03.03.X1 and 03.02.0R2 (the Riemannian curvature specialisations). Single-paragraph edit.
Pointer in 05.02.06-geodesic-flow-hamiltonian.md to 03.03.X1 (the Riemannian geodesic equation that is the projection of the Hamiltonian flow). Single-paragraph edit.
Pointer in 06.04.05-dbar-hilbert-pde.md to 03.02.0R11 (the higher-dim Riemannian analogue of the Riemann-surface case). Single-paragraph edit.
Cross-link in 07.04.07-riemannian-symmetric-space.md (when that unit ships from the Helgason batch) to Jost Ch. 6 as the Riemannian-geometric secondary anchor.

§4 Implementation sketch (P3 → P4)

For a full Jost coverage pass, priority-1 closes the basic-Riemannian gap shared with Milnor / Helgason / KN-I; priority-2 opens the geometric-analysis chapter that has no other Codex anchor; priority-3 is the Yamabe capstone. Realistic production estimate (mirroring earlier Milnor / KN-I / Brown / Bott-Tu batches):

~3 hours per Riemannian-geometry unit; ~4 hours per geometric- analysis unit because the analytic chapters require careful Sobolev / PDE notation and the Eells-Sampson and Yamabe proofs are substantial.
Priority 1: 6 units, with 03.03.X1 and 03.02.0R3 at ~3000– 2000 words and the rest at ~1500. Estimate ~20 hours.
Priority 2: 6 units, with 03.02.0R8 at ~3000 words. Estimate ~22 hours.
Priority 3: 2 units, with 03.02.0R13 at ~3500 words. Estimate ~9 hours.
Priority 4: 1 Master unit + 5 weaving edits. Estimate ~3 hours.
Total for full coverage: ~54 hours, roughly two focused weeks. Priority-1 alone is ~20 hours and closes the audit-chain Riemannian gap from Cycles 3, 4, and 5 in a single batch.

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

Bernhard Riemann, "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (Habilitationsvortrag, Göttingen, 10 June 1854; published 1868 in Abh. Königl. Ges. Wiss. Göttingen 13) — the birth of Riemannian geometry; the metric tensor as a positive- definite quadratic form on tangent vectors. Cite in 03.02.0R1.
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 parallel-transport / Levi-Civita- connection definition. Cite in 03.03.X1.
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 (shared with KN-I punch-list).
Élie Cartan, "Sur certaines formes riemanniennes remarquables des géométries à groupe fondamental simple," Annales scientifiques de l'É.N.S. 44 (1927) 345–467 (and the 1925–26 papers cited in Milnor MMT) — Cartan-Hadamard. Jacques Hadamard, "Les surfaces à courbures opposées et leurs lignes géodésiques," J. Math. Pures Appl. 4 (1898) 27–73 — the original negative-curvature case. Cite in 03.02.0R4.
Sumner B. Myers, "Riemannian manifolds with positive mean curvature," Duke Math. J. 8 (1941) 401–404 — Bonnet-Myers. Cite in 03.02.0R4.
Harry Ernest Rauch, "A contribution to differential geometry in the large," Annals of Math. 54 (1951) 38–55 — Rauch comparison. Cite in 03.02.0R5.
Victor A. Toponogov, "Riemann spaces with curvature bounded below" (Russian), Uspekhi Mat. Nauk 14 (1959) 87–130 — Toponogov triangle comparison. Cite in 03.02.0R5.
James Eells, Joseph H. Sampson, "Harmonic mappings of Riemannian manifolds," American J. Math. 86 (1964) 109–160 — the founding paper of harmonic-map theory; harmonic-map heat flow; existence into nonpositively curved targets. Cite in 03.02.0R6, 03.02.0R7, 03.02.0R8. Top-tier originator citation — no other Codex unit currently anchors this paper.
Hidehiko Yamabe, "On a deformation of Riemannian structures on compact manifolds," Osaka Math. J. 12 (1960) 21–37 — the Yamabe-problem statement and the (flawed) original proof attempt. Cite in 03.02.0R13.
Neil S. Trudinger, "Remarks concerning the conformal deformation of Riemannian structures on compact manifolds," Annali Scuola Norm. Sup. Pisa 22 (1968) 265–274 — identification of the gap in Yamabe's proof; low-dimensional case. Cite in 03.02.0R13.
Thierry Aubin, "Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire," J. Math. Pures Appl. 55 (1976) 269–296 — Yamabe in dim $\geq 6$ , non-locally-conformally-flat. Cite in 03.02.0R13.
Richard Schoen, "Conformal deformation of a Riemannian metric to constant scalar curvature," J. Diff. Geom. 20 (1984) 479–495 — completion of the Yamabe proof using the positive mass theorem. Cite in 03.02.0R13.
Jürgen Jost (2017, 7th ed.) — the canonical consolidation. Cite throughout as the in-Codex master anchor.

Notation crosswalk. Jost writes:

$g_{ij}$ and $g^{ij}$ for the metric and its inverse in coordinates, $g = g_{ij} d x^{i} \otimes d x^{j}$ .
$Γ_{ij}^{k}$ for Christoffel symbols (lower indices symmetric); $\nabla_{X} Y$ for the Levi-Civita covariant derivative.
$R (X, Y) Z$ for the curvature tensor; $R_{ijkl} = g(R(\partial_i, \partial_j) \partial_k, \partial_l)$ with Jost's sign convention (the convention agreeing with do Carmo and Lee, opposite to Cheeger-Ebin and Petersen).
$K (σ)$ for sectional curvature of a 2-plane $σ$ ; $Ric$ for Ricci; $Scal$ (sometimes $R$ or $S$ ) for scalar curvature.
$τ (u)$ for the tension field of a map; $E (u)$ for the energy.
$W^{k, p} (M)$ for Sobolev spaces.

The Codex notation decision (per docs/specs/UNIT_SPEC.md §11): adopt Jost's sign convention for $R$ (already aligned with Lee and do Carmo) and Jost's $τ / E$ for harmonic maps. Use $Scal$ for scalar curvature consistently (not $R$ , which collides with the curvature tensor). Record in a §Notation paragraph of 03.03.X1, 03.02.0R2, and 03.02.0R6.

§5 What this plan does NOT cover

A line-number-precise inventory of every named theorem in Jost. The book is 697 pp.; without a local PDF the audit relies on Jost's chapter-level structure as documented in the Springer ToC and in Aubin / Hebey / Lee-Parker citations. A full P1 audit at line-number granularity is deferred to a dedicated pass once a copy is on hand.
Ricci flow. Jost Ch. 11–12 sets the analytic foundation but does not cover the Hamilton-Perelman programme. Ricci flow is a separate Fast Track audit anchored on Topping Lectures on the Ricci Flow, Chow-Knopf The Ricci Flow: An Introduction, or Morgan-Tian Ricci Flow and the Poincaré Conjecture.
Geometric measure theory. Federer / Simon / Almgren are Fast Track 3.x entries handled in a separate audit; Jost touches the topic only in passing.
Lorentzian geometry. Jost is strictly Riemannian (positive definite). The Lorentzian / pseudo-Riemannian story is in O'Neill Semi-Riemannian Geometry, Sternberg Semi-Riemannian Geometry (local copy at reference/fasttrack-texts/01-fundamentals/), or Beem-Ehrlich-Easley Global Lorentzian Geometry.
Symplectic and Kähler geometry. Already covered in 05-symplectic/ and partial Kähler coverage in 06-riemann-surfaces/.
Spin and Dirac. Already covered in 03.09-spin-geometry/ via Lawson-Michelsohn; Jost Ch. 7 is the lighter parallel exposition and adds no new units.

§6 Acceptance criteria for FT equivalence (Jost)

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

All six priority-1 units have shipped. This closes the basic-Riemannian gap shared with Milnor (Cycle 3), KN-I (Cycle 4), and Helgason (Cycle 5) audits in a single batch.
≥4 of the six priority-2 units have shipped (03.02.0R6, 03.02.0R7, 03.02.0R8, plus at least one of the Sobolev / embedding / regularity trio).
Both priority-3 Yamabe units have shipped, OR the Yamabe units are explicitly deferred to a follow-up batch with a Pass-W pointer in place.
≥95% of Jost's named theorems in Chs. 1–5 and 9–12 map to Codex units. Currently ~5%; after priority-1+2+3 this rises to ~85%; full ≥95% requires the priority-4 deepenings.
≥90% of Jost's worked computations in Chs. 9–12 (harmonic-map examples, Yamabe test-function calculations) have a direct unit or are referenced from a unit that covers them.
Notation decisions are recorded (see §4).
Pass-W weaving connects the new units to 03.02-manifolds/, 03.05-bundles/, 02.11-functional-analysis/, 03.09-spin-geometry/ (via 03.09.17), 05.02-hamiltonian/ (via 05.02.06), and 06.04-cohomology/ (via 06.04.05).

The 6 priority-1 units close the audit-chain gap. Priority-2 closes the harmonic-map and Sobolev gap. Priority-3 closes the Yamabe gap. Priority-4 is optional deepening + weaving.

§7 Sourcing

Not free. Springer commercial title, Universitext series. No author-hosted PDF.
Buy. ISBN 978-3-319-61859-3 (7th ed. 2017, paperback). Anna's Archive carries scanned copies of the 6th and 7th editions for research access (clearly marked as the fallback per the standing acquisition policy in this plan series).
Local copy. When acquired, add to reference/fasttrack-texts/03-modern-geometry/ as Jost-RiemannianGeometryAndGeometricAnalysis.pdf to mirror the pattern of other commercial FT texts (Sternberg, Bott, May).
Secondary anchors (already in the citation chain above).
- do Carmo, Riemannian Geometry, Birkhäuser 1992. The first-course standard. BUY.
- Petersen, Riemannian Geometry, GTM 171, Springer 3rd ed. 2016. The strongest comparison-geometry alternative. BUY.
- Lee, Introduction to Riemannian Manifolds, GTM 176, Springer 2nd ed. 2018. The cleanest modern exposition for priority-1 material. BUY.
- Cheeger-Ebin, Comparison Theorems in Riemannian Geometry, AMS Chelsea 2008. The deeper comparison-geometry reference. BUY.
- Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer
  1. The geometric-analysis companion for priority-2 and -3 material. BUY.
- Hebey, Sobolev Spaces on Riemannian Manifolds, LNM 1635, Springer 1996. The Sobolev-on-manifolds reference. BUY.
- Lee-Parker, "The Yamabe Problem," Bull. AMS 17 (1987) 37–91. Free at AMS. The canonical Yamabe survey. Cite alongside Jost Ch. 12.

Jost — *Riemannian Geometry and Geometric Analysis* (Fast Track 3.28) — Audit + Gap Plan