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Riemannian Metric
Graduate+
Definition
Structure assigning inner product on tangent vectors at each point
Formulas
g: Tₚ M × Tₚ M → ℝ
ds² = g_ij dxⁱ dx^j
Length: L(γ) = ∫√(g_ij γ̇ⁱ γ̇^j) dt
Examples
Example 1
Euclidean metric: g_{ij} = δ_{ij}
Example 2
Sphere metric: ds² = dθ² + sin²θ dφ²
Applications
General relativity
Geodesics
Curvature
Related Documents
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