Riemannian Metrics

Differential Geometry · Axiom Academy

Understanding how to equip manifolds with a notion of distance, angle, and curvature 1. Definition of a Riemannian Metric Definition: Let M be a smooth manifold. A Riemannian metric g on M is a smooth assignment of an inner product g p to each tangent space T p M at every point p ∈ M . More precisely, for each point p ∈ M , the metric g p is a bilinear form: This assignment must vary smoothly with the base point p . In local coordinates, if X and Y are vector fields, then g ( X , Y ) is a smooth function. 2. Properties of a Riemannian Metric For a Riemannian metric g p : T p M × T p M → ℝ, the following three properties must hold at each point p : Bilinearity: For all v , w , u ∈ T p M and a , b ∈ ℝ: Symmetry: For all v , w ∈ T p M : Positive Definiteness: For all non-zero v ∈ T p M : These properties ensure that g p behaves like a genuine inner product at each tangent space, allowing us to define lengths and angles. 3. From Manifold to Riemannian Manifold The addition of a Riemannian metric fundamentally changes the nature of a manifold: Definition: A Riemannian manifold is a pair ( M , g ) where M is a smooth manifold and g is a Riemannian metric on M . Once we have a Riemannian metric, we can define: Length of tangent vectors: The norm of v ∈ T p M is given by: Angle between vectors: For non-zero vectors v , w ∈ T p M : Length of curves: For a smooth curve γ : [ a , b ] → M :

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