If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line ( i. e. a geodesic which minimizes distance on each interval ) then it is isometric to a direct product of the real line and a complete ( n-1 )- dimensional Riemannian manifold which has nonnegative Ricci curvature.

Important examples of such techniques include classical multidimensional scaling ( which is identical to PCA ), Isomap ( which uses geodesic distances in the data space ), diffusion maps ( which uses diffusion distances in the data space ), t-SNE ( which minimizes the divergence between distributions over pairs of points ), and curvilinear component analysis.

This observation has resulted in practical applications in chemical and processing industry: in order to minimize the change in entropy of a system, one should follow the minimum geodesic path between the desired endpoints of the process.

) The notion of geodesic curvature allows to distinguish the part of the curvature in ambient space that is due to the submanifold ( the normal curvature ) and the one that comes from the curve itself.

If the Cauchy surface were compact, i. e. space is compact, the null geodesic generators of the boundary can intersect everywhere because they can intersect on the other side of space.

His first " continuous tension – discontinuous compression " geodesic dome ( full sphere in this case ) was constructed at the University of Oregon Architecture School in 1959 with the help of students.

* LBNL – Led the construction of the geodesic structure that holds the PMTs

The analysis of the radial geodesic motion of a massive particle into an Einstein – Rosen bridge shows that the proper time of the particle extends to infinity.

The Earth is not a perfect sphere, it is an oblate spheroid or ellipsoid ( i. e. – slightly compressed at the poles ), which means that the shortest distance between two points ( a geodesic ) is not quite a great circle.

* 1937 – Albert Einstein, Leopold Infeld, and Banesh Hoffmann show that the geodesic equations of general relativity can be deduced from its field equations

Having become assistant to Carlos Guillermo Moesta ( 1825 – 1884 ), director of the observatory at Santiago de Chile, in 1859, he was associated with the Chilean geodesic survey in 1864.

These decks will house twelve geodesic dome-shaped ' pods ' – semi permanent tent structures-and a separate shower and toilet cubicle for each.

is sometimes called the energy or action of the curve ; this name is justified because the geodesic equations are the Euler – Lagrange equations of motion for this action.

* June 29 – Buckminster Fuller is granted a United States patent for his development of the geodesic dome.

* 1967 – Expo 67 in Montreal features the American pavilion, a geodesic dome designed by Buckminster Fuller, and the Habitat 67 housing complex designed by Moshe Safdie.

In mathematics, the Hopf – Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds.

The observatory participated in the basic geodesic work, namely in measuring degrees of the arc of the meridian from the Danube to the Arctic Ocean ( until 1851 ), and in triangulation of Spitsbergen in 1899 – 1901.

Because of the very simple structure of this Hamiltonian, the equations of motion for the particle, the Hamilton – Jacobi equations, are nothing other than the geodesic equations on the manifold: the particle moves along geodesics.

Gago Coutinho ( 1869 – 1959 ), officer of the Portuguese Navy, navigator and historian, headed a geodesic mission to São Tomé between 1915 and 1918 when marks were place as a basis for a geodetic network in the archipelago, and after that observations for triangulation, precise base measurement and astronomical observations were made.

* 1968: Brandon Carter solves the geodesic equations for Kerr – Newmann electrovacuum,

For affine buildings, this metric satisfies the CAT ( 0 ) comparison inequality of Alexandrov, known in this setting as the Bruhat-Tits non-positive curvature condition for geodesic triangles: the distance from a vertex to the midpoint of the opposite side is no greater than the distance in the corresponding Euclidean triangle with the same side-lengths ( see ).

He also developed numerous inventions, mainly architectural designs, the best known of which is the geodesic dome.

One reason for this given by the researchers is that the B-80 is actually more like the original geodesic dome structure popularized by Buckminster Fuller, which uses triangles rather than hexagons.

This new class of preferred motions, too, defines a geometry of space and time — in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential.

This and related predictions follow from the fact that light follows what is called a light-like or null geodesic — a generalization of the straight lines along which light travels in classical physics.

In relativity theory, orbits follow geodesic trajectories which approximate very well to the Newtonian predictions.

The points which are furthest separated in longitude are apart along a geodesic line.

Physically, these describe different universes in which all the same events and interactions are still ( causally ) possible, but a new additional force is necessary to effect this ( that is, replication of all the same trajectories would necessitate departures from geodesic motion because the metric is different ).

Each of the edges must be prevented from leaking, which can be quite challenging for a geodesic structure.

In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer.

Conversely, a non-compact geometry can have closed geodesics: on an infinite cylinder, which is a non-compact flat geometry, a loop around the cylinder is a closed geodesic.

In this case, for two events which are simultaneous according the cosmological time coordinate, the value of the cosmological proper distance is not equal to the value of the proper length between these same events ,( Wright ) which would just be the distance along a straight line between the events in a Minkowski diagram ( and a straight line is a geodesic in flat Minkowski spacetime ), or the coordinate distance between the events in the inertial frame where they are simultaneous.

There are experiments using geodesic grids and icosahedral grids, which ( being more uniform ) do not have pole-problems.

We can also reparametrize geodesics to be unit speed, so equivalently we can define exp < sub > p </ sub >( v ) = β (| v |) where β is the unit-speed geodesic ( geodesic parameterized by arc length ) going in the direction of v. As we vary the tangent vector v we will get, when applying exp < sub > p </ sub >, different points on M which are within some distance from the base point p — this is perhaps one of the most concrete ways of demonstrating that the tangent space to a manifold is a kind of " linearization " of the manifold.

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space.

The formula above might not be true for points which are not close enough since the geodesic might for example wrap around the manifold ( e. g. on a sphere ).

Specifically, the scalar curvature represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space.

The condition of positive Ricci curvature is most conveniently stated in the following way: for every geodesic there is a nearby initially parallel geodesic which will bend toward it when extended, and the two will intersect at some finite length.

Presumably, at the end of the geodesic the observer has fallen into a singularity or encountered some other pathology at which the laws of general relativity break down.

These theorems, strictly speaking, prove that there is at least one non-spacelike geodesic that is only finitely extendible into the past but there are cases in which the conditions of these theorems obtain in such a way that all past-directed spacetime paths terminate at a singularity.

Like Newton's first law of motion, Einstein's theory states that if a force is applied on an object, it would deviate from a geodesic.

The geodesic curvature of geodesics being zero, and the Euler characteristic of T being 1, the theorem then states that the sum of the turning angles of the geodesic triangle is equal to 2π minus the total curvature within the triangle.

# The Cheeger-Gromoll splitting theorem states that if a complete Riemannian manifold with contains a line, meaning a geodesic γ such that for all, then it is isometric to a product space.