Riemannian Metric

★★★★★Graduate+

📖Definition

Structure assigning inner product on tangent vectors at each point

g: Tₚ M × Tₚ M → ℝ

ds² = g_ij dxⁱ dx^j

Length: L(γ) = ∫√(g_ij γ̇ⁱ γ̇^j) dt

예제 1

Euclidean metric: g_{ij} = δ_{ij}

예제 2

Sphere metric: ds² = dθ² + sin²θ dφ²

General relativity

Geodesics

Curvature

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