![]() ![]() As a result, such a manifold is necessarily a ( pseudo-) Riemannian manifold. An invariant metric implies that the structure group of the frame bundle is the orthogonal group O( p, q). Abstractly, one would say that the manifold has an associated ( orthonormal) frame bundle, with each " frame" being a possible choice of a coordinate frame. ![]() However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. also do not require the concept of a metric. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. Array of numbers describing a metric connection
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