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The geodesic flow defines a family of curves in the tangent bundle. The derivatives of these curves define a vector field on the total space of the tangent bundle, known as the geodesic spray. More precisely, an affine connection gives rise to a splitting of the double tangent bundle TTM into horizontal and vertical bundles:
There are several ways of defining geodesics (Hilbert & Cohn-Vossen 1952, pp. 220–221).A simple definition is as the shortest path between two points on a surface. However, it is frequently more useful to define them as paths with zero geodesic curvature—i.e., the analogue of straight lines on a curved su
The full geodesic equation is + = where s is a scalar parameter of motion (e.g. the proper time), and are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) symmetric in the two lower indices.
An affine connection is a rule which describes how to legitimately move a vector along a curve on the manifold without changing its direction. By definition, an affine connection is a bilinear map Γ ( T M ) × Γ ( T M ) → Γ ( T M ) {\displaystyle \Gamma (TM)\times \Gamma (TM)\to \Gamma (TM)} , where Γ ( T M ) {\displaystyle \Gamma (TM ...
The Riemannian manifold with its Levi-Civita connection is geodesically complete if the domain of every maximal geodesic is (,). [25] The plane R 2 {\displaystyle \mathbb {R} ^{2}} is geodesically complete.
where D denotes the covariant derivative with respect to the Levi-Civita connection, R the Riemann curvature tensor, ˙ = / the tangent vector field, and t is the parameter of the geodesic. On a complete Riemannian manifold, for any Jacobi field there is a family of geodesics γ τ {\displaystyle \gamma _{\tau }} describing the field (as in the ...
Geodesic of a photon emitted from a light source located on the event horizon of a black hole, with an impact parameter = = and then moving to the unstable orbit =. If b < b c r i t {\displaystyle b<b_{crit}} the photon is released towards infinity.
A section of the rear fuselage from a Vickers Warwick showing the geodetic construction in duralumin. On exhibit at the Armstrong & Aviation Museum at Bamburgh Castle.. A geodetic airframe is a type of construction for the airframes of aircraft developed by British aeronautical engineer Barnes Wallis in the 1930s (who sometimes spelt it "geodesic").