Differential Geometry

Contents

Differentiable Manifold

Tangent Vector

Tangent Space

Exterior Derivative

k-form φ = g dxⁱ = g dx^i₁ ∧ ⋯ ∧ dx^i_k. dφ = D_x^j g x^j ∧ xⁱ. General k-form ω = f_i dxⁱ. dφ = d(dxⁱ) = ... General k-form dω = D_x^j f_j dx^j ∧ xⁱ. For a 1-form ω(dω)_{i, j} = D_iω_j - D_jω_i. dxⁱ D_xⁱ = 1. dω(x, y) = d_x(ω(y)) - d_y(ω(x)) - ω([x, y]).

Covariant Derivative

Riemannian Geometry

Connection

Affine Connection

Levi-Civita Connection

Metric Connection

Symplectic Geometry

Tensor Density

Pseudo-Riemannian Manifold

Derivation

Differential Form

D(a b) = a D(b) + D(a) b.

Laplace–Beltrami Operator

Hodge Theory

ω = dα + δβ + γ, Δλ = 0. Ω_k(M) ≅ im d_{k - 1} ⊕ im δ_{k + 1} ⊕ H_Δ^k(M).

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.