The Metric Tensor

Differential Geometry · Axiom Academy

A rigorous introduction to the fundamental structure that defines distance and angles on manifolds 1. The Metric Tensor as a (0,2)-Tensor Field Symmetric: g(X,Y) = g(Y,X) for all vector fields X, Y Non-degenerate: If g(X,Y) = 0 for all Y, then X = 0 Positive definite: g(X,X) > 0 for all nonzero X As a (0,2)-tensor, the metric takes two vector fields and produces a scalar function. At each point p ∈ M, it defines an inner product on the tangent space T p M. 2. Components g ij in Local Coordinates In a local coordinate system (x 1 , x 2 , ..., x n ), the metric tensor has components defined by: where ∂ i = ∂/∂x i are the coordinate basis vectors. The metric tensor can then be written as: where dx i ⊗ dx j denotes the tensor product of coordinate differentials. For any two tangent vectors: 3. The Metric Matrix [g ij ] and Its Properties The components g ij form a matrix at each point: Symmetry: g ij = g ji (only n(n+1)/2 independent components) Positive Definite: For any nonzero vector v i : g ij v i v j > 0 Smoothly Varying: Each g ij is a smooth function of the coordinates Non-singular: det[g ij ] > 0 everywhere 4. Transformation Law Under Coordinate Changes Under a coordinate transformation x i → x̄ k , the metric components transform as a (0,2)-tensor: This is the defining property of a covariant 2-tensor. The transformation ensures that the metric represents an intrinsic geometric object independent of coordinates.

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