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differential geometers suck at naming things

alex pawełko

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“Differential geometry is the subject of things invariant under change-of-notation”

-Apocryphal, popularized by John M. Lee

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we’re really bad at picking names

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we’re really bad at picking names

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we’re really bad at picking names

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we’re really bad at picking names

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we’re really bad at picking names

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we’re really bad at picking names

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we’re really bad at picking names

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we’re really bad at picking names

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we’re really bad at picking names

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we’re really bad at picking names

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we’re really bad at picking names

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Theorem (Berger’s Holonomy Theorem, 1955)

Let (M, g) be an irreducible nonsymmetric Riemannian manifold. Then Hol(g) is one of the following groups:

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group

name

dimension

SO(n)

generic case (oriented Riemannian)

any

U(m)

Kaehler (torsion-free Hermitian)

n = 2m is even

SU(m)

​

Calabi-Yau (torsion-free complex-oriented Hermitian)

n = 2m is even

Sp(m)

hyper-Kaehler (torsion-free quaternionic Hermitian)

n = 4m

Sp(m)Sp(1)

Quaternionic Kaehler

n = 4m

G2

(torsion-free) G2

n = 7

Spin(7)

(torsion-free) Spin(7)

n = 8

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group

name

dimension

SO(n)

generic case (oriented Riemannian)

any

U(m)

Kaehler (torsion-free Hermitian)

n = 2m is even

SU(m)

​

Calabi-Yau (torsion-free complex-oriented Hermitian)

n = 2m is even

Sp(m)

hyper-Kaehler (torsion-free quaternionic Hermitian)

n = 4m

Sp(m)Sp(1)

Quaternionic Kaehler

n = 4m

G2

(torsion-free) G2

n = 7

Spin(7)

(torsion-free) Spin(7)

n = 8

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group

name

dimension

SO(n)

generic case (oriented Riemannian)

any

U(m)

Kaehler (torsion-free Hermitian)

n = 2m is even

SU(m)

​

Calabi-Yau (torsion-free complex-oriented Hermitian)

n = 2m is even

Sp(m)

hyper-Kaehler (torsion-free quaternionic Hermitian)

n = 4m

Sp(m)Sp(1)

Quaternionic Kaehler

n = 4m

G2

(torsion-free) G2

n = 7

Spin(7)

(torsion-free) Spin(7)

n = 8

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group

name

dimension

SO(n)

generic case (oriented Riemannian)

any

U(m)

Kaehler (torsion-free Hermitian)

n = 2m is even

SU(m)

​

Calabi-Yau (torsion-free complex-oriented Hermitian)

n = 2m is even

Sp(m)

hyper-Kaehler (torsion-free quaternionic Hermitian)

n = 4m

Sp(m)Sp(1)

Quaternionic Kaehler

n = 4m

G2

(torsion-free) G2

n = 7

Spin(7)

(torsion-free) Spin(7)

n = 8

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group

name

dimension

SO(n)

generic case (oriented Riemannian)

any

U(m)

Kaehler (torsion-free Hermitian)

n = 2m is even

SU(m)

​

Calabi-Yau (torsion-free complex-oriented Hermitian)

n = 2m is even

Sp(m)

hyper-Kaehler (torsion-free quaternionic Hermitian)

n = 4m

Sp(m)Sp(1)

Quaternionic Kaehler

n = 4m

G2

(torsion-free) G2

n = 7

Spin(7)

(torsion-free) Spin(7)

n = 8

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group

name

dimension

SO(n)

generic case (oriented Riemannian)

any

U(m)

Kaehler (torsion-free Hermitian)

n = 2m is even

SU(m)

​

Calabi-Yau (torsion-free complex-oriented Hermitian)

n = 2m is even

Sp(m)

hyper-Kaehler (torsion-free quaternionic Hermitian)

n = 4m

Sp(m)Sp(1)

Quaternionic Kaehler

n = 4m

G2

(torsion-free) G2

n = 7

Spin(7)

(torsion-free) Spin(7)

n = 8

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group

name

dimension

SO(n)

generic case (oriented Riemannian)

any

U(m)

Kaehler (torsion-free Hermitian)

n = 2m is even

SU(m)

​

Calabi-Yau (torsion-free complex-oriented Hermitian)

n = 2m is even

Sp(m)

hyper-Kaehler (torsion-free quaternionic Hermitian)

n = 4m

Sp(m)Sp(1)

Quaternionic Kaehler

n = 4m

G2

(torsion-free) G2

n = 7

Spin(7)

(torsion-free) Spin(7)

n = 8

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these pde are really totally ass

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Let’s look at U(m)-structures:

By the holonomy principle, get linear algebraic objects:�

g, J, ω, NJ, θ

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Name

PDE

Kaehler

​

Nearly Kaehler

​

Almost Kaehler

​

Hermitian semi-Kaehler

​

Locally conformally Kaehler

​

quasi-Kaehler

​

Hermitian

​

coalmost Kaehler

​

semi-Kaehler

​

Locally conformally almost Kaehler

​

​

​

​

​

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Name

PDE

Kaehler

​

Nearly Kaehler

​

Almost Kaehler

​

Hermitian semi-Kaehler

​

Locally conformally Kaehler

​

quasi-Kaehler

​

Hermitian

​

coalmost Kaehler

​

semi-Kaehler

​

Locally conformally almost Kaehler

​

​

​

​

​

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Name

PDE

Kaehler

​

Nearly Kaehler

​

Almost Kaehler

​

Hermitian semi-Kaehler

​

Locally conformally Kaehler

​

quasi-Kaehler

​

Hermitian

​

coalmost Kaehler

​

semi-Kaehler

​

Locally conformally almost Kaehler

​

​

​

​

​

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Name

PDE

Kaehler

​

Nearly Kaehler

​

Almost Kaehler

​

Hermitian semi-Kaehler

​

Locally conformally Kaehler

​

quasi-Kaehler

​

Hermitian

​

coalmost Kaehler

​

semi-Kaehler

​

Locally conformally almost Kaehler

​

​

​

​

​

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Name

PDE

Kaehler

​

Nearly Kaehler

​

Almost Kaehler

​

Hermitian semi-Kaehler

​

Locally conformally Kaehler

​

quasi-Kaehler

​

Hermitian

​

coalmost Kaehler

​

semi-Kaehler

​

Locally conformally almost Kaehler

​

​

​

​

​

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Name

PDE

Kaehler

​

Nearly Kaehler

​

Almost Kaehler

​

Hermitian semi-Kaehler

​

Locally conformally Kaehler

​

quasi-Kaehler

​

Hermitian

​

coalmost Kaehler

​

semi-Kaehler

​

Locally conformally almost Kaehler

​

​

​

​

​

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Name

PDE

Kaehler

​

Nearly Kaehler

​

Almost Kaehler

​

Hermitian semi-Kaehler

​

Locally conformally Kaehler

​

quasi-Kaehler

​

Hermitian

​

coalmost Kaehler

​

semi-Kaehler

​

Locally conformally almost Kaehler

​

​

​

​

​

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Name

PDE

Kaehler

​

Nearly Kaehler

​

Almost Kaehler

​

Hermitian semi-Kaehler

​

Locally conformally Kaehler

​

quasi-Kaehler

​

Hermitian

​

coalmost Kaehler

​

semi-Kaehler

​

Locally conformally almost Kaehler

​

​

​

​

​

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Name

PDE

Kaehler

​

Nearly Kaehler

​

Almost Kaehler

​

Hermitian semi-Kaehler

​

Locally conformally Kaehler

​

quasi-Kaehler

​

Hermitian

​

coalmost Kaehler

​

semi-Kaehler

​

Locally conformally almost Kaehler

​

​

​

​

​

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Name

PDE

Kaehler

​

Nearly Kaehler

​

Almost Kaehler

​

Hermitian semi-Kaehler

​

Locally conformally Kaehler

​

quasi-Kaehler

​

Hermitian

​

coalmost Kaehler

​

semi-Kaehler

​

Locally conformally almost Kaehler

​

​

​

​

​

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There are more… (whose PDE are really annoying to write down)

pseudo-Kaehler

paraKaehler

nearly pseudo-Kaehler

almost pseudo-Kaehler (this is the same PDE as almost Kaehler with an extra linear algebraic condition)

nearly paraKaehler

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And the ultimate paper

(which i actually read)

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You understand everything in this equation�(there are 3 implicit sums, 6 implicit derivatives, and many many omitted isomorphisms we are assuming are =)

Gamma^k_{ij}d/dx^k = \nab_i d/dx^j