Knowra Metric tensor Metric tensor A symmetric tensor field that assigns an inner product to tangent vectors at each point of a manifold. It defines lengths, angles, volumes, and, through shortest paths, distances.
Tangent space : A vector space of possible tangent directions at a point on a manifold. The metric assigns an inner product to vectors in each tangent space.
Tensor field : A smoothly varying assignment of tensors to points of a manifold. A metric tensor is a rank-two covariant tensor field with additional properties.
Pseudo-Riemannian metric : A nondegenerate symmetric metric that may assign positive, negative, or zero squared lengths to nonzero vectors. Unlike a Riemannian metric, it need not be positive definite.
Gaussian curvature : An intrinsic measure of curvature at a point on a two-dimensional surface. For surfaces, the metric determines Gaussian curvature through its derivatives.
Carl Friedrich Gauss : A German mathematician whose work established foundational results in number theory, geometry, and physics. His intrinsic study of surface geometry helped motivate metric-based geometry.
Inner product : A bilinear operation on a vector space that encodes lengths and angles. At each point, the metric is an inner product on the tangent space.
Covariant tensor : A multilinear function of vector inputs, represented by covariant indices. A metric takes two tangent vectors as inputs and returns a scalar.
Lorentzian metric : A pseudo-Riemannian metric with one direction of one sign and all remaining directions of the other. This indefinite metric models spacetime in general relativity.
Scalar curvature : A scalar obtained by contracting the curvature tensor of a metric. It summarizes one aspect of the metric’s curvature at each point.
Bernhard Riemann : A German mathematician whose work transformed geometry, analysis, and number theory. His 1854 lecture introduced the broader framework of geometry defined by a metric.
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