Differential Geometry
Contents
Differentiable Manifold
Tangent Vector
Tangent Space
Exterior Derivative
k-form φ = g dxi = g dxi1 ∧ ⋯ ∧ dxik. dφ = Dxj g xj ∧ xi. General k-form ω = fi dxi. dφ = d
Covariant Derivative
Riemannian Geometry
Connection
Affine Connection
Levi-Civita Connection
Metric Connection
Symplectic Geometry
Tensor Density
Pseudo-Riemannian Manifold
Derivation
Differential Form
D
Laplace–Beltrami Operator
Hodge Theory
ω = dα + δβ + γ, Δλ = 0. Ωk
Complex Differential Form
Lie Group
Continuous Symmetry
Lie Derivative
References
- Sergio Fabi. An Invitation to Synthetic Differential Geometry. 2017.
- Anders Kock. New Methods for Old Spaces: Synthetic Differential Geometry. 2017.
- Anders Kock. Synthetic Geometry of Manifolds. 2010.
- Tim de Laat. Synthetic Differential Geometry: An application to Einstein’s Equivalence Principle. 2008.
- Anders Kock. Synthetic Differential Geometry. 2006.