Length and Angles

Differential Geometry · Axiom Academy

LESSON Length and Angle on Manifolds Understanding metric structures: how the Riemannian metric enables measurement of lengths, angles, areas, and volumes on curved spaces This generalizes the Euclidean norm ||v|| = √(v · v). The metric tensor g acts as an inner product on the tangent space T p M, making it a Euclidean vector space at each point. This is the Cauchy-Schwarz inequality applied to the metric inner product. The inequality |g(v,w)| ≤ ||v|| ||w|| ensures that cos θ ∈ [-1, 1], making the angle well-defined. Here γ'(t) = dγ/dt is the tangent vector to the curve. This integral accumulates the infinitesimal lengths ||γ'(t)|| dt along the curve. where g is the matrix representation of the metric tensor in coordinates (u, v). The determinant measures how the metric "stretches" infinitesimal coordinate rectangles. 5. Volume Elements on n-Manifolds This is the natural measure on M induced by the metric. It allows integration of functions: ∫ M f dV = ∫ M f√(det g) dx 1 ...dx n 6. Orthonormal Frames and Orthogonality Here δ ij is the Kronecker delta. This means ||e i || = 1 and e i ⊥ e j for i ≠ j. 7. Computing Arc Length on Surfaces Write the metric in coordinates: g = E du 2 + 2F du dv + G dv 2 Compute the tangent vector: γ'(t) = (u'(t), v'(t)) Evaluate: g(γ', γ') = E(u') 2 + 2F u'v' + G(v') 2 Integrate: L = ∫ a b √(g(γ', γ')) dt

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