This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian and pseudo-Riemannian manifolds.
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In a Riemannian manifold M with metric tensor g, the length of a continuously differentiable curve γ : a,b → M is defined by
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Then into the 1940s and 1950s a certain sub-genre of St. Louis modernism emerged, with the locally important Harris Armstrong, and a series of daring modern civic landmarks like Gyo Obata's Planetarium, the geodesic-dome Climatron, and the main terminal building at Lambert-St. Louis International Airport.
Located inside a large geodesic sphere structure, similar to Spaceship Earth at Walt Disney World's Epcot.
To zeroth order, the path of the lighter object will be a geodesic in the Kerr spacetime of the supermassive black hole.
A geodesic (or geodetic) airframe is a type of construction for the airframes of aircraft developed by British aeronautical engineer Barnes Wallis in the 1930s.
Atul Chitnis, Senior VP, GeoDesic Bangalore - 1985 Batch of Mechanical Engineering
He organized conferences which included such notables as Edward Teller, inventor of the American hydrogen bomb, and Buckminster Fuller, inventor of the geodesic dome.
As a first lieutenant in the USAF, Bull was stationed at MIT Lincoln Laboratory and worked with Buckminster Fuller on developing the geodesic radar domes for the Distant Early Warning Line (DEW Line) system at the north slope of Alaska.
Douglas Trumbull, director of the 1972 science fiction classic film Silent Running, stated that the geodesic domes on the spaceship Valley Forge were based on the Missouri Botanical Garden's Climatron dome.
The quantum ergodicity theorem of Shnirelman, Yves Colin de Verdière, and Zelditch states that a compact Riemannian manifold whose unit tangent bundle is ergodic under the geodesic flow is also ergodic in the sense that the probability density associated to the nth eigenfunction of the Laplacian tends weakly to the uniform distribution on the unit cotangent bundle as n → ∞.
Since Ricci curvature measures the amount by which the volume of a small geodesic ball deviates from the volume of a ball in Euclidean space, small geodesic balls will have no volume deviation, but their "shape" may vary from the shape of the standard ball in Euclidean space.
triangulated manifolds (together with James Sethian), the geodesic active contours algorithm for image segmentation, a geometric framework for image filtering
Lasenby, Doran, and Gull of Cambridge University have proposed a new formulation of gravity, termed gauge theory gravity (GTG), wherein spacetime algebra is used to induce curvature on Minkowski space while admitting a gauge symmetry under "arbitrary smooth remapping of events onto spacetime" (Lasenby, et al.); a nontrivial proof then leads to the geodesic equation,