Holonomy and Calibrated Geometry
for the Pure Mathematician and Interested Physicist

These are the notes I made as I taught myself about holonomy, in preparation for my DPhil after the summer. Whilst useful for myself to check my understanding by the “Feynman technique”, I also thought others may find use in it.

These are a research-oriented introduction to calibrated geometry: closed differential forms that bound volume from above and pick out submanifolds which are automatically volume-minimising. The point is simple and powerful: if a form \(\varphi\) calibrates a submanifold \(L\), then \(L\) is minimal without solving Euler–Lagrange equations.

Our path runs from Harvey–Lawson’s foundational theory to the calibrated geometries arising from special holonomy: Kähler and Calabi–Yau (with complex and special Lagrangian calibrations), and the exceptional realms of \(G_{2}\) and \(\mathrm{Spin}(7)\) (associatives/coassociatives and Cayley cycles). Along the way we touch moduli, deformation theory, and gluing.

Primary references: Harvey–Lawson, Calibrated Geometries (Acta Math. 1982); Joyce, Riemannian Holonomy and Calibrated Geometry; Joyce, Compact Manifolds with Special Holonomy; and expository notes by McLean and Salur on deformations of calibrated submanifolds.

Course Schedule

Lecture I: A Prelude and Introduction to Calibrated Geometry

A web of geometry · Minimality · Calibrations · Calibrated submanifolds · Basic properties

Minimal submanifolds are of great interest: a Riemannian submanifold is a critical point of the volume functional iff its mean curvature vanishes. Yet critical needn’t mean minimising. Can we identify submanifolds that minimise volume within their homology class? Calibrated submanifolds do...automatically! We explore first examples and properties.

Calibration Volume Bound Euclidean Examples
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Lecture II — Holonomy: The Geometric Engine

Connections · Parallel Transport

Connections and parallel transport on vector/principal bundles. Holonomy and restricted holonomy. Ambrose–Singer: curvature generates holonomy. Berger’s list in outline to motivate special metrics.

Holonomy Ambrose–Singer Berger Overview
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Lecture III — The Bridge: Special Holonomy & Calibrated Geometry

CY · \(G_{2}\) · \(\mathrm{Spin}(7)\)

How exceptional holonomy furnishes calibrations: complex/Kähler (complex submanifolds), Calabi–Yau (special Lagrangians), \(G_{2}\) (associatives/coassociatives), \(\mathrm{Spin}(7)\) (Cayley). Minimality and deformation theory in each setting.

SLag Associative/Coassociative McLean
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