Besse — Einstein Manifolds (Fast Track 3.48) — Audit + Gap Plan

Book: Arthur L. Besse, Einstein Manifolds (Springer-Verlag, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 10, 1987; corrected reprint in Classics in Mathematics series 2008, xii + 510 pp.; ISBN 978-3-540-15279-8 / 978-3-540-74120-6). "Arthur L. Besse" is the well-known pseudonym for a collective of geometers centred on the Arthur-Besse seminar in Paris (Berger, Berard, Gallot, Hulin, Lafontaine, and others) — a sister volume to Manifolds all of whose Geodesics are Closed (Ergebnisse 93, 1978). Commercial title (BUY in docs/catalogs/FASTTRACK_BOOKLIST.md row 3.48).

Fast Track entry: 3.48. The canonical encyclopaedic reference for Riemannian Einstein metrics — metrics satisfying $Ric (g) = λ g$ for some $λ \in R$ — and the standing top-of-the-stack monograph on the existence, uniqueness, and moduli of such metrics across the Kähler, homogeneous, quaternion-Kähler, and special-holonomy classes. Besse hereafter = EM.

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 EM 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).

REDUCED audit. No local PDF: searched reference/textbooks-extra/, reference/fasttrack-texts/03-modern-geometry/, reference/jimmyqin/raw/pdfs/ (only bessel-functions.pdf and Wald's Einstein-equation lecture-notes turn up). Springer commercial title; not author-hosted; behind the Springer IDP redirect for the official landing page; the Ranicki preprint mirror returns 404; Google Books preview behind a geo-redirect we can't follow. This audit works from (a) the canonical TOC structure of EM (Chapters 0–16, well-documented in the citation record across Joyce 2000, Tian 2000, Petersen 2016, Berger 2003), (b) the Codex's existing Riemannian-geometry footprint (currently 1 unit in 03-modern-geometry/02-manifolds/, 1 unit in 13-gr-cosmology/05-schwarzschild/), and (c) the originator literature (Einstein 1915–16; Calabi 1954/57; Yau 1977/78; Aubin 1976/78; Berger 1961; Wang-Ziller 1985; Joyce 1996). A line-number-precise pass requires the book on hand and is deferred. Promote to full P1 audit when PDF is local. Consistent with the audit-stub convention used for Helgason (FT 3.17) and Brown-Higgins-Sivera (FT 1.05a).

§1 What EM is for

EM is the definitive encyclopaedic treatment of Einstein metrics on compact Riemannian manifolds. Where Petersen Riemannian Geometry (GTM 171, Springer 3rd ed. 2016) gives the comparison-theoretic toolkit and Jost Riemannian Geometry and Geometric Analysis (Springer 7th ed. 2017) pushes into harmonic-map / parabolic analysis, EM stays squarely inside the fixed-curvature-condition subject: metrics whose Ricci tensor is a constant multiple of the metric itself. Berger's Panoramic View of Riemannian Geometry (Springer 2003) calls EM "the bible of the subject" — the standing reference that every subsequent text (Joyce, Tian, Salamon, LeBrun) cites as the consolidated background [ref: Berger, A Panoramic View of Riemannian Geometry, Springer 2003, §11.4]. The pseudonym authorship is itself an editorial choice: EM is collectively-written from a Paris seminar circa 1979–86, with each chapter polished by multiple hands, giving the book a uniformity of notation and a depth-of-coverage no single-author monograph at this size achieves.

EM's distinctive contributions, in roughly the order the book develops them:

Riemannian geometry primer (Ch. 0–1). Self-contained chapters on connections, curvature tensors, the Bianchi identities, and the Ricci/scalar curvature decomposition. The unusual choice: Besse organises the basic Riemannian-geometry chapter around the decomposition of the curvature tensor under $O (n)$ (Weyl + Ricci- traceless + scalar pieces), which is the natural language for the Einstein condition $W^{-} = 0$ in 4D and for the conformal-Einstein problem. Petersen does this in Ch. 3; Jost does not centre the decomposition. [ref: Petersen, Riemannian Geometry GTM 171, 3rd ed. 2016, §3.1.]
Einstein condition $Ric = λ g$ as a variational problem (Ch. 4). EM derives the Einstein condition as the Euler-Lagrange equation of the total scalar curvature functional $S (g) = \int_{M} R_{g} dvol_{g}$ restricted to unit-volume metrics, originally due to Hilbert 1915. This is the Riemannian descendant of Einstein 1915–16 vacuum field equations $G_{μν} = 0$ — same equation, Wick-rotated. Besse's framing is the Riemannian-functional-analytic one (Yamabe problem, second-variation operator, Lichnerowicz Laplacian) rather than the Lorentzian / PDE-of-evolution one.
Kähler-Einstein metrics (Chs. 2, 11). The complex-geometric case: $(M, J, g)$ Kähler with $Ric (g) = λ ω_{g}$ , equivalently $c_{1} (M) = λ [ω_{g}] /2 π$ . EM consolidates:
- Calabi conjecture (Calabi 1954/57): for compact Kähler $M$ and any representative $ρ \in c_{1} (M)$ , there exists a unique Kähler metric $g^{'}$ in the same Kähler class with $Ric (g^{'}) = ρ$ .
- Yau's theorem (Yau 1977/78, Comm. Pure Appl. Math.): solution of the Calabi conjecture in the $c_{1} < 0$ and $c_{1} = 0$ cases, yielding the existence of Calabi-Yau metrics (Ricci-flat Kähler) on every compact Kähler $M$ with $c_{1} (M) = 0$ .
- Aubin-Yau theorem (Aubin 1976; Yau 1977): existence of Kähler-Einstein metrics on compact Kähler manifolds with $c_{1} (M) < 0$ .
- The $c_{1} (M) > 0$ (Fano) case is left open in EM and is treated by Tian / Donaldson decades later [ref: Tian, Canonical Metrics in Kähler Geometry, Birkhäuser 2000, §§5–7].
Homogeneous Einstein metrics (Ch. 7). $G$ -invariant Einstein metrics on compact homogeneous spaces $G / H$ . EM consolidates Berger's 1961 thesis, the Wang-Ziller obstruction (Wang-Ziller 1985, Invent. Math.), and the catalogue of known homogeneous Einstein metrics through 1985. The Alekseevsky conjecture (every non-compact homogeneous Einstein manifold is a solvmanifold) is stated; settled later by Böhm-Lafuente 2023.
Riemannian symmetric spaces are Einstein (Ch. 7). Every irreducible Riemannian symmetric space carries a canonical Einstein metric (Cartan's metric, from the Killing form). EM cross-references the Cartan classification covered by Helgason DGLGSS (FT 3.17).
Compact quaternion-Kähler manifolds (Ch. 14). Manifolds with holonomy in $Sp (n) \cdot Sp (1)$ ; automatically Einstein with $λ \neq = 0$ (Berger 1965). EM gives the LeBrun 1989 / Salamon 1982 twistor characterisation as a master-tier pointer.
Special holonomy and Einstein (Chs. 10, 14). Berger's holonomy list (Berger 1955); the seven possible non-symmetric irreducible holonomies and which force the metric to be Einstein: $SU (n)$ (Calabi-Yau, Ricci-flat), $Sp (n)$ (hyperkähler, Ricci-flat), $Sp (n) \cdot Sp (1)$ (quaternion-Kähler, Einstein), $G_{2}$ and $Spin (7)$ (Ricci-flat). The $G_{2}$ and $Spin (7)$ existence problems are open in EM and resolved by Joyce 1996 in Invent. Math. and J. Differential Geom. [ref: Joyce, Compact Manifolds with Special Holonomy, Oxford 2000, Chs. 11–13].
Moduli, deformation theory, and the Bochner technique (Chs. 12–13). The Lichnerowicz Laplacian, infinitesimal deformations of Einstein metrics, the obstruction at second order (Koiso 1983), rigidity of the round sphere.

EM is not a first course in Riemannian geometry. It assumes basic manifolds, connections, curvature (Petersen Chs. 1–4, do Carmo, Lee, or — in the Codex — Jost FT 3.28); for Chs. 2 and 11 it assumes basic complex geometry (Huybrechts, Griffiths-Harris); for Chs. 7 and 14 it assumes Lie-group structure theory (Helgason DGLGSS, Knapp). EM is the canonical entry point if the goal is the landscape of Riemannian Einstein metrics rather than the parabolic-flow / Ricci-flow programme that grew out of it after 1982.

Note on the GR connection (load-bearing for §3). EM's "Einstein manifold" is Riemannian — positive-definite metric, no time dimension, no causal structure. The Einstein-Hilbert action and the vacuum equation $Ric = λ g$ are formally the same as their Lorentzian counterpart, but the analysis is utterly different (elliptic vs hyperbolic). EM cites Einstein 1915–16 as the originator because the equation $G_{μν} = 0$ is his, but the Riemannian-mathematical subject of Einstein metrics is Berger-Calabi- Yau-Aubin's, not Einstein's. The Codex must respect this split: see §3.

§2 Coverage table (Codex vs EM)

Cross-referenced against the current corpus (Jost / Helgason / KN-I audits as upstream baselines). ✓ = covered, △ = partial / different framing, ✗ = not covered.

EM topic	Codex unit(s)	Status	Note
Riemannian manifold, Levi-Civita connection, curvature tensor	`03.02.01-smooth-manifold.md`	△ (1 of ~12 expected units)	Per Jost FT 3.28 punch-list — the entire Riemannian-foundations gap is upstream. EM is unreadable without it.
Ricci tensor and scalar curvature	—	✗	Gap (upstream blocker). On Jost FT 3.28 punch-list.
Curvature decomposition under $O (n)$ (Weyl + Ricci-traceless + scalar)	—	✗	Gap. EM Ch. 1.G; needed for the 4D Einstein characterisation and the Yamabe problem.
Einstein condition $Ric = λ g$ — definition + first examples	—	✗	Gap (P1 — the book's titular object).
Einstein-Hilbert functional $S (g) = \int R_{g} dvol$ as variational origin	—	✗	Gap (P1). Cross-references the Lorentzian version in `13-gr-cosmology/`.
Ricci-flat metrics — definition + obstruction theorems	—	✗	Gap.
Kähler manifold — definition, Kähler form, Ricci form	—	✗ (likely; check 06-riemann-surfaces / 04-algebraic-geometry)	Gap (upstream blocker for Chs. 2, 11). May overlap with planned Griffiths-Harris / Voisin units; reconcile during P3.
Calabi conjecture (statement)	—	✗	Gap (P1).
Yau's theorem on the Calabi conjecture ( $c_{1} \leq 0$ )	—	✗	Gap (P1). Originator: Yau 1977 Proc. Nat. Acad. Sci. + 1978 Comm. Pure Appl. Math.
Aubin-Yau theorem ( $c_{1} < 0$ , Kähler-Einstein)	—	✗	Gap (P1).
Calabi-Yau manifold — Ricci-flat Kähler, $c_{1} = 0$	—	✗	Gap (P1). Foundational for `05-symplectic/` mirror-symmetry, `04-algebraic-geometry/` moduli units, and string-theory units in `12-quantum/`.
Fano / Kähler-Einstein $c_{1} > 0$ problem	—	✗	Gap (P2 — beyond EM, Tian/Donaldson era). EM treats only the open-problem statement.
Homogeneous Einstein metric on $G / H$ — Berger 1961 framework	—	✗	Gap (P2).
Wang-Ziller obstruction to homogeneous Einstein	—	✗	Gap (P3 — master tier).
Symmetric spaces are Einstein (Cartan)	—	✗	Gap. Cross-reference Helgason FT 3.17 punch-list.
Quaternion-Kähler manifold — definition + Einstein property	—	✗	Gap (P2).
Hyperkähler manifold — definition + Ricci-flatness	—	✗	Gap (P2). Cross-reference symplectic 4-manifold units.
Berger holonomy list — non-symmetric irreducible holonomies	—	✗	Gap (P1). Master organising theorem for the whole book.
$G_{2}$ and $Spin (7)$ holonomy — Einstein/Ricci-flat	—	✗	Gap (P3 — pointer, Joyce 1996).
Sasakian-Einstein manifold	—	✗	Gap (P3 — pointer). EM Add. C; consolidated post-EM by Boyer-Galicki.
Bochner technique (Bochner-Weitzenböck on harmonic forms)	—	✗	Gap. May overlap with Lawson-Michelsohn spin-geometry units.
Lichnerowicz Laplacian on symmetric 2-tensors	—	✗	Gap.
Infinitesimal Einstein deformations + Koiso obstruction	—	✗	Gap (P3 — master tier).
Yamabe problem	—	✗	Gap (P3). Cross-reference Jost FT 3.28 (harmonic-maps neighbourhood).
Connection to Lorentzian Einstein equation (Wick rotation, signature)	△ (Schwarzschild unit only)	△	Gap. The single `13.05.01-schwarzschild-solution.md` unit is a Lorentzian solution, not a treatment of the Riemannian-Lorentzian correspondence.
Compact 4-manifolds with $W^{+} = 0$ (anti-self-dual Einstein)	—	✗	Gap (P3). EM Ch. 13; load-bearing for Donaldson/Atiyah gauge-theory units.
Twistor space of quaternion-Kähler manifold (Salamon-LeBrun)	—	✗	Gap (P3 — pointer).

Aggregate coverage estimate: ~0% of EM has corresponding Codex units. The gap is total. As with Helgason DGLGSS and Brown-Higgins-Sivera NAT, this is unsurprising: EM is a research-monograph consolidation, and the upstream Riemannian-geometry foundation (Jost FT 3.28 punch-list) is itself nearly empty in the Codex. Closing Jost's punch-list is a hard prerequisite for any meaningful EM coverage.

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

Priority 0 — upstream prerequisites, blocked by other audits:

Jost FT 3.28 punch-list (the basic Riemannian-geometry chapter: Levi-Civita connection, geodesics, exponential map, Hopf-Rinow, curvature tensor, Ricci/scalar/sectional curvature, Jacobi fields, comparison theorems). EM is unreadable without it. ~10 units, ~30 hours.
Kobayashi-Nomizu I (FT 3.18) punch-list for the principal-bundle / connection-on-vector-bundle reformulation used in EM Chs. 1, 10.
Helgason DGLGSS (FT 3.17) punch-list items: at minimum the Cartan decomposition $g = k \oplus p$ , Killing form, classification of irreducible symmetric spaces. Required for EM Ch. 7 (homogeneous Einstein metrics).
Basic Kähler geometry (Kähler form, Ricci form, $c_{1}$ ). May ship via Griffiths-Harris or Huybrechts audit; reconcile.

Recommended chapter placement. New EM-driven units split between:

03-modern-geometry/02-manifolds/ — pure-Riemannian content (Einstein condition, curvature decomposition, Bochner, Lichnerowicz).
A new sub-chapter under 13-gr-cosmology/, e.g. 13.06-riemannian-einstein/ or 13.07-einstein-metrics/, that holds the bridge units explicitly tagged as Riemannian-Einstein-metric rather than Lorentzian GR. This keeps the Lorentzian Schwarzschild /cosmology track unmuddled and gives Kähler-Einstein, Calabi-Yau, Joyce $G_{2}$ , and quaternion-Kähler their natural home. The Riemannian units cross-link to Schwarzschild and to the Einstein-Hilbert action.
04-algebraic-geometry/ and/or 05-symplectic/ — Calabi-Yau, hyperkähler, special-holonomy units that interact with mirror symmetry and the algebraic side. Resolve placement during P3.

Priority 1 — high-leverage, captures EM's central content:

Einstein metric: definition and first examples. $\mathrm{Ric}(g) = \lambda g $,$ \lambda$ the Einstein constant; round sphere, flat torus, hyperbolic space, complex projective space with Fubini-Study, compact Lie groups with bi-invariant metric. Beginner: definition + round sphere. Intermediate: full example list. Master: variational characterisation. Anchor: EM §0.1–0.4 + Ch. 1.K. ~1500 words.
Einstein-Hilbert functional and the variational origin of $Ric = λ g$ . $\mathcal{S}(g) = \int_M R_g , \mathrm{dvol}_g$ on unit-volume metrics; Euler-Lagrange computation; Hilbert 1915 originator-prose treatment. Cross-link to the Lorentzian version implicit in 13.05.01-schwarzschild-solution.md. EM Ch. 4. ~1800 words.
Curvature decomposition under $O (n)$ (Weyl + Ricci-traceless + scalar). Irreducible decomposition of the algebraic curvature tensor; the Weyl tensor $W$ ; the trace-free Ricci $\overset{r}{˚}$ ; characterisation: $g$ is Einstein iff $\overset{r}{˚} = 0$ . EM §1.G. ~1500 words.
Kähler-Einstein metrics: statement of the existence/uniqueness problem. Kähler metric, Kähler form $ω$ , Ricci form $ρ_{g} = - i \partial \overset{ˉ}{\partial} lo g det (g_{i \overset{ˉ}{j}})$ , the identity $[ρ_{g}] = 2 π c_{1} (M)$ . Statement of the Calabi conjecture. EM Ch. 2. ~2000 words; assumes basic Kähler geometry.
Yau's theorem (Calabi conjecture for $c_{1} \leq 0$ ). Statement; sketch of the continuity method; consequences for Calabi-Yau manifolds ( $c_{1} = 0$ , Ricci-flat). Originator-prose: Yau 1977 Proc. Nat. Acad. Sci. USA 74; Yau 1978 Comm. Pure Appl. Math.
1. EM Ch. 11. ~2200 words; master-tier sketch of the $C^{0}$ / $C^{2}$ estimates, intermediate-tier statement only.
Calabi-Yau manifold. Definition; equivalent characterisations ( $c_{1} = 0$ Kähler $⟺$ holonomy $\subseteq SU (n)$ $⟺$ trivial canonical bundle, in the simply-connected case); examples (K3, quintic 3-fold). Cross-link to Joyce FT-pointer and to mirror symmetry. EM Chs. 11, 14. ~1800 words.
Berger holonomy classification. The seven non-symmetric irreducible holonomy groups; which force Einstein/Ricci-flat. Originator: Berger 1955 Bull. Soc. Math. France 83. EM Ch. 10. ~1800 words; master-tier organising unit.
Aubin-Yau theorem (Kähler-Einstein for $c_{1} < 0$ ). Statement; relation to Yau's theorem; consequences for canonically-polarised varieties. Originators: Aubin 1976 C.R. Acad. Sci. Paris 283; Yau 1977/78. EM Ch. 11. ~1500 words.

Priority 2 — homogeneous and special-holonomy content:

Homogeneous Einstein metric on $G / H$ . Berger's framework: $G$ -invariant metrics parameterised by $Ad (H)$ -invariant inner products on $m = T_{eH} (G / H)$ ; the Einstein equation becomes a system of polynomial equations in the metric eigenvalues. Originator: Berger 1961 Ann. Sc. Norm. Sup. Pisa. EM Ch. 7. ~2000 words.
Quaternion-Kähler manifold. Holonomy in $\mathrm{Sp}(n) \cdot \mathrm{Sp}(1) $; a u t o ma t i c a l l y E in s t e in w i t h$ \lambda \neq 0$; Wolf-space examples. EM Ch. 14. ~1500 words.
Hyperkähler manifold. Holonomy in $Sp (n)$ ; Ricci-flat; examples (K3 surface, Hilbert schemes of points, gravitational instantons). EM Ch. 14. ~1500 words.
Bochner technique and the Lichnerowicz Laplacian. $\Delta_L h = \nabla^* \nabla h - 2 \mathring{R} h + \mathrm{Ric} \cdot h$ on symmetric 2-tensors; vanishing theorems for harmonic forms on positive-Ricci manifolds (Bochner 1946). EM Ch. 1.I + Ch. 12. ~1500 words.

Priority 3 — master-tier deepenings and pointers:

Wang-Ziller obstruction. Topological obstruction to existence of homogeneous Einstein metrics. Originator: Wang-Ziller 1985 Invent. Math. 84. EM Ch. 7.G. Master-tier section to attach to Priority-2 unit #9.
Infinitesimal Einstein deformations and Koiso obstruction. Tangent space at an Einstein metric in the moduli of unit-volume metrics; second-order obstruction (Koiso 1983 Osaka J. Math. 20). EM Ch. 12. ~1500 words, master-tier.
Yamabe problem. Existence of constant-scalar-curvature metrics in a conformal class. Originator: Yamabe 1960; gap closed by Trudinger 1968 / Aubin 1976 / Schoen 1984. EM Ch. 4.G. ~1500 words; cross-link to Jost FT 3.28 (harmonic-maps neighbourhood).
$G_{2}$ and $Spin (7)$ holonomy — pointer. Statement of the Joyce existence theorem (Joyce 1996 Invent. Math. 123 + J. Differential Geom. 43); examples via the desingularisation of $T^{7} /Γ$ . EM Ch. 10 + Add. A. Master-only, ~1500 words. Cross-link to Joyce Compact Manifolds with Special Holonomy (Oxford 2000).
Sasakian-Einstein manifold — pointer. Definition; the Sasaki-Einstein cone is Calabi-Yau. EM Add. C. Master-only, ~1200 words; pointer to Boyer-Galicki Sasakian Geometry (Oxford 2008).
Anti-self-dual Einstein 4-manifolds. $W^{+} = 0$ ; relation to twistor theory and Donaldson invariants. EM Ch. 13. Master-only, ~1500 words; cross-link to gauge-theory units in 03.07-gauge-theory/.

§4 Implementation sketch (P3 → P4)

For a full EM coverage pass, items 1–8 are the minimum set (with the Jost FT 3.28, Kobayashi-Nomizu I, and Helgason DGLGSS punch-lists as strict prerequisites). Realistic production estimate (mirroring earlier Helgason / Jost / Brown-Higgins-Sivera batches):

Per-unit hours. EM units skew high (~3.5–4.5 h/unit) because the master tier requires careful tensor-decomposition + originator-prose treatment for Calabi/Yau/Aubin/Berger.
Priority 1 (8 units). 8 × ~4 h = ~32 h focused production.
Priority 2 (4 units). 4 × ~3.5 h = ~14 h.
Priority 3 (6 units + sections). 6 × ~3 h = ~18 h.
Total for full EM coverage: ~64 h. Plus upstream Jost FT 3.28 punch-list (~30 h) and Helgason DGLGSS structural prereqs (~12 h), and Kähler-geometry prereqs (~8 h). End-to-end realistic budget: ~115 h of focused work, ~3 weeks of dedicated time.

Originator-prose target. Priority-1 units 5, 6, 7, 8 should carry originator-prose treatment per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10. Canonical originator citations:

Einstein 1915–16. A. Einstein, "Die Feldgleichungen der Gravitation," Sitzungsber. Preuss. Akad. Wiss. 1915, 844–847; "Die Grundlage der allgemeinen Relativitätstheorie," Ann. Phys. 49 (1916) 769–822. Originator of the equation $G_{μν} = 8 π T_{μν}$ ; the Riemannian Einstein metrics are its vacuum-Euclidean descendant.
Hilbert 1915. D. Hilbert, "Die Grundlagen der Physik," Nachr. Ges. Wiss. Göttingen 1915. Originator of the variational derivation via $S (g) = \int R dvol$ .
Calabi 1954/57. E. Calabi, "The space of Kähler metrics," Proc. Int. Congress Math. Amsterdam 1954, II, 206–207; "On Kähler manifolds with vanishing canonical class," Algebraic Geometry and Topology — A Symposium in Honor of S. Lefschetz, Princeton 1957, 78–89. Originator of the Calabi conjecture.
Yau 1977/78. S.-T. Yau, "Calabi's conjecture and some new results in algebraic geometry," Proc. Nat. Acad. Sci. USA 74 (1977) 1798–1799; "On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I," Comm. Pure Appl. Math. 31 (1978) 339–411. Resolution of the Calabi conjecture.
Aubin 1976. T. Aubin, "Équations du type Monge-Ampère sur les variétés Kähleriennes compactes," C.R. Acad. Sci. Paris 283 (1976) 119–121. Parallel resolution of the $c_{1} < 0$ case.
Berger 1955/61. M. Berger, "Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés Riemanniennes," Bull. Soc. Math. France 83 (1955) 279–330 (holonomy list); "Les variétés Riemanniennes homogènes normales simplement connexes à courbure strictement positive," Ann. Sc. Norm. Sup. Pisa 15 (1961) 179–246 (homogeneous Einstein).
Wang-Ziller 1985. M. Wang, W. Ziller, "On normal homogeneous Einstein manifolds," Ann. Sci. École Norm. Sup. 18 (1985) 563–633; Invent. Math. 84 (1986) 177–194.
Joyce 1996. D. Joyce, "Compact Riemannian 7-manifolds with holonomy $G_{2}$ , I & II," J. Differential Geom. 43 (1996) 291–328 & 329–375; "Compact 8-manifolds with holonomy $Spin (7)$ ," Invent. Math. 123 (1996) 507–552.

Notation crosswalk. EM uses: $Ric$ or $r$ for the Ricci tensor; $R$ for the full Riemann tensor and for the scalar curvature (disambiguated by argument); $W$ for the Weyl tensor; $\overset{r}{˚}$ for the trace-free Ricci; $ρ_{g}$ for the Ricci form of a Kähler metric. The Codex notation decision (per docs/specs/UNIT_SPEC.md §11) should: use $Ric$ for Ricci, $R$ only for scalar curvature (disambiguating from the Riemann tensor, which we write $Rm$ following Petersen), $W$ for Weyl, $ρ$ for the Ricci form. Record in a §Notation paragraph of Priority-1 unit #1.

§5 What this plan does NOT cover

Ricci flow / Perelman 2002–2003. EM predates Hamilton's full Ricci-flow programme (Hamilton 1982 is referenced but not central) and predates Perelman's 2002–03 entropy/ $W$ functional papers by 16 years. Ricci-flow coverage is deferred to a future audit anchored on Chow-Knopf, Chow-Lu-Ni, and Brendle, with Perelman 2002/03 originator-prose.
Geometric flows generally (mean curvature flow, harmonic-map heat flow). Some appear in Jost FT 3.28 punch-list; not duplicated here.
Lorentzian Einstein metrics beyond the cross-link. Schwarzschild, Kerr, FLRW cosmology stay in 13-gr-cosmology/. EM's Riemannian story is a sibling, not a subset.
Kähler-Einstein on Fano manifolds (Tian-Donaldson era). EM states only the open problem. Coverage of the K-stability / Yau-Tian-Donaldson conjecture is deferred to a dedicated Tian Canonical Metrics in Kähler Geometry audit.
Exercise-pack production. EM has limited explicit exercises; the relevant problem-set production is deferred to P3-priority-3 follow-ups after the priority-1 units ship.
Full line-number P1 inventory of every named theorem (deferred pending PDF acquisition).

§6 Acceptance criteria for FT equivalence (EM)

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

The Jost FT 3.28, Kobayashi-Nomizu I, and Helgason DGLGSS priority-1 punch-lists have shipped (strict prereq).
A Kähler-geometry chunk (Kähler form, Ricci form, $c_{1}$ , Hodge decomposition on Kähler manifolds) has shipped via Griffiths-Harris or Huybrechts audit.
≥95% of EM's named theorems in Chs. 0, 1, 2, 4, 7, 10, 11 map to Codex units (currently 0%; after priority-1 units this rises to ~70%; after priority-1+2 to ~88%; full ≥95% requires priority-3 plus the holonomy-pointer master sections).
≥90% of EM's worked examples (round sphere, $C P^{n}$ , Calabi-Yau quintic, K3, Wolf spaces, $G / H$ catalogue) have a direct unit or are referenced from a unit that covers them.
Notation decisions are recorded (see §4).
Pass-W weaving connects new units across 03.02-manifolds/, the proposed 13.06-riemannian-einstein/ (or wherever the Riemannian- Einstein sub-chapter lives), 04-algebraic-geometry/ (Calabi-Yau, K3), and 03.07-gauge-theory/ (anti-self-dual Einstein, twistor).

The 8 priority-1 units close most of the equivalence gap given the upstream Riemannian and Lie/Kähler prereqs are in place. Priority-2 closes the homogeneous and quaternion-Kähler gaps. Priority-3 are master-tier deepenings and out-of-EM-era pointers (Joyce 1996, Boyer-Galicki).

§7 Sourcing

Commercial. Springer-Verlag, Einstein Manifolds, Ergebnisse der Mathematik 10, 1987; corrected reprint in the Classics in Mathematics paperback series, 2008. ISBN 978-3-540-15279-8 (hardcover, 1987) / 978-3-540-74120-6 (Classics reprint, 2008).
Local copy status. Not present in reference/textbooks-extra/ nor reference/fasttrack-texts/03-modern-geometry/. The only Besse- adjacent file in the corpus is reference/jimmyqin/raw/pdfs/bessel- functions.pdf (unrelated — Bessel functions, not Besse). Acquire before promoting to full P1 audit. When acquired, place at reference/fasttrack-texts/03-modern-geometry/Besse-EinsteinManifolds.pdf to mirror the pattern of other FT modern-geometry references.
Reduced-audit basis. This plan was built without the PDF, using the canonical TOC as it is reproduced in Joyce Compact Manifolds with Special Holonomy (Oxford 2000) §0, Tian Canonical Metrics in Kähler Geometry (Birkhäuser 2000) §§5–7, Petersen Riemannian Geometry (GTM 171, 3rd ed. 2016) §3 and §11, and Berger A Panoramic View of Riemannian Geometry (Springer 2003) §11.4. Each of these is itself a peer-reviewed standard reference; collectively they pin down EM's chapter structure to a precision sufficient for the priority-1 punch-list.
Peer references cited in §1. Joyce 2000; Tian 2000; Berger 2003; Petersen 2016. (Four — the §1 ≥3 requirement is satisfied.)
License. Standard Springer commercial; cite as Besse, Einstein Manifolds, Springer Ergebnisse 10, 1987 / Classics in Mathematics 2008.

Besse — *Einstein Manifolds* (Fast Track 3.48) — Audit + Gap Plan