"Geodesic distance" redirects here. For distances on the surface of a sphere, see Great-circle distance. For distances on the surface of the Earth, see Geodesics on an ellipsoid. For the edge-count of a shortest path in a graph, see Distance (graph theory).
In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to (or very close to) that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the lengths of all paths from the first point to the second. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space.
If the space has the stronger property that there always exists a path that achieves the infimum of length (a geodesic) then it is called a geodesic metric space or geodesic space. For instance, the Euclidean plane is a geodesic space, with line segments as its geodesics. The Euclidean plane with the origin removed is not geodesic, but is still a length metric space.
a metric space relative to the intrinsicmetric is defined as the infimum of the lengths of all paths from the first point to the second. A metric space...
}}\right|_{\tau =0}.} Jordan curve Killing vector field Length metric the same as intrinsicmetric. Levi-Civita connection is a natural way to differentiate...
(typically non-Minkowski) norm on each tangent space of M. The induced intrinsicmetric dL: M × M → [0, ∞] of the original quasimetric can be recovered from...
the metric d is intrinsic) if the distance between any two points x and y is the infimum of lengths of paths between them. Unlike in a geodesic metric space...
statements about the geodesic completeness of Riemannian manifolds Intrinsicmetric – Concept in geometry/topology Isotropic line Jacobi field Morse theory –...
mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by...
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface)...
Banach fixed-point theorem Polish space Hausdorff distance Intrinsicmetric Category of metric spaces Stone duality Stone's representation theorem for Boolean...
consider Lipschitz Riemannian metrics or measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to...
itself, is determined by the intrinsicmetric of the surface without any further reference to the ambient space: it is an intrinsic invariant. In particular...
gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion, so it does not mean that the universe expands "into" anything...
ISBN 978-0-07-048293-7. T Yanao; K Takatsuka (2005). "Effects of an intrinsicmetric of molecular internal space". In Mikito Toda; Tamiki Komatsuzaki; Stuart...
the gravitational constant. The Schwarzschild metric has a singularity for r = 0, which is an intrinsic curvature singularity. It also seems to have a...
which is not metrically convex. Intrinsicmetric Khamsi, Mohamed A.; Kirk, William A. (2001). An Introduction to Metric Spaces and Fixed Point Theory....
S^{2}(1)} is a model for the two-dimensional elliptic plane. It carries an intrinsicmetric that arises by measuring geodesic length. The geodesic circles are...
In acoustics and fluid dynamics, an acoustic metric (also known as a sonic metric) is a metric that describes the signal-carrying properties of a given...
The intrinsic dimension for a data set can be thought of as the number of variables needed in a minimal representation of the data. Similarly, in signal...
the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component. The name is motivated by the importance...
differential geometry in higher dimensions. This intrinsic point of view in terms of the Riemannian metric, denoted by d s 2 {\displaystyle ds^{2}} by Riemann...
Curvature of Riemannian manifolds of dimension at least two can be defined intrinsically without reference to a larger space. For curves, the canonical example...
2173896 and 2233796 is 3. For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as...
of Met as a category; they may also be defined intrinsically in terms of a Helly property of their metric balls, and because of this alternative definition...
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