Geometry MOC

Differential geometry MOC

Differential geometry studies smooth geometric structures, in particular differentiable manifolds. As such it is closely related to differential topology. A guiding principle is the Linearization dogma.

In an ambient space

John Willard Milnor and Lyle Noakes take an approach where only real embedded manifolds are considered, as justified by the Whitney embedding theorem.

• Differentiability on arbitrary subsets of real coördinate space
• Tangent space
• Differential pushforward

In these notes lowercase Latin indices are used for abstract index notation unless otherwise specified, see the linked Zettel for information on conventions.

Spaces

• Submanifold
• Tangent bundle
• Vector bundle

Morphisms

• Immersion and submersion

Fibrations

• Bundle map
• Bundle section
• Vector bundle morphism

Derived maps

• Tangent map
• Differential pushforward
• Differential pullback

Results

• Inverse function theorem gives a condition for local diffeomorphisms
• Preïmage theorem can be used to verify most constructions that are manifolds

Attached data

• Tangent space, Tangent bundle
• Cotangent space, Cotangent bundle

Calculus on manifolds

Besides from -morphisms, other “functions” we might want to differentiate include

• Scalar field, Vector field, 1-form
  • Commuting vector fields are frames
• Tensor field
• Differential form
  • Volume form, Riemannian volume form

This gives rise to a plethora of derivative notions, which all agree when they act on a scalar field :

• Exterior derivative of a differential form.
• Lie derivative of a tensor field along a vector field.
• Covariant derivative of a tensor field (requires an affine connexion).
• Differential pushforward and differential pullback of a -morphism.

Some pain has been taken to make sure most of my definitions for Calculus of variations MOC work for manifolds.

Differentiation

There are three main definitions of differentiation, which in some sense coïncide on scalar fields.

• Exterior derivative
• Covariant derivative via an Affine connexion
• Lie derivative

Geometry

• Smooth geodesic

Symmetry

• Diffeomorphism
• Differential pullback along a diffeomorphism defines symmetries of tensor fields and in particular metrics
• Isometry of a semi-Riemannian manifold
• Killing field

Extra structure

• Semi-Riemannian manifold
  • Vielbein
• Oriented manifold

Examples

• Spherical coördinates

---

develop | en | sembr