In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, you can follow a "straight" line indefinitely along any direction. More formally, the exponential map at point p, is defined on T_pM, the entire tangent space at p.

Is sphere Geodesically complete?

The Euclidean spaces ℝn and the (round or squashed) n-spheres Sn are geodesically complete. But any open ball of finite radius inside ℝn is not.

What does Geodesically mean?

1. geodesical - of or relating to or determined by geodesy. geodesic, geodetic.

Is geodesic smooth?

A broken geodesic is a piecewise smooth curve as in Def- inition 7.1. 1, where each curve segment is a geodesic. Proposition 7.2. 11 A Riemannian manifold, (M,g), is connected iff any two points of M can be joined by a broken geodesic.

What is geodesic problem?

The geodesic problem on a triaxial ellipsoid is solved as a boundary value problem, using the calculus of variations. ... From the solution, the ellipsoidal coordinates and the angle between the line of constant longitude and the geodesic, at any point along the geodesic, are determined.

Thomas Leistner, Geodesic completeness of compact Lorentzian manifolds

34 related questions found

What is the difference between geodetic and geodesic?

There is a substantial difference between the two: Geodesy is basically geographical surveying and measurement, often at a large scale and including longitude and latitude issues, while a Geodesic is about extending some properties of straight lines to curved and other spaces.

What is a complete Riemannian manifold?

In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, you can follow a "straight" line indefinitely along any direction. More formally, the exponential map at point p, is defined on T_pM, the entire tangent space at p.

Are manifolds compact?

-dimensional sphere and torus. Compact manifolds in two dimensions are completely classified by their orientation and the number of holes (genus). When there is need for a separate term, a compact boundaryless manifold is called a closed manifold. ...

What does geodesic mean in English?

: the shortest line between two points that lies in a given surface.

What is geodesic line?

geodesic line - (mathematics) the shortest line between two points on a mathematically defined surface (as a straight line on a plane or an arc of a great circle on a sphere)

What is a geodesic in spacetime?

In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.

What is geodesic software?

Geodesic is an innovator in software products focused on Information, Communication and Entertainment for mobile phones and desktop computers. Headquartered in Mumbai, India, Geodesic's Mundu suite of award-winning products includes solutions for Instant Messaging, Voice-over-IP and Internet Radio.

What is geodesic app?

At its core, Geodesic is a way to “like” something in the real word. Find a great restaurant or a hidden gem in your city, let everyone know.

What language does geodesic come from?

geodesy (n.) 1560s, "the art of land surveying," from Modern Latin geodaesia, from Greek geodaisia "division of the earth;" ultimately from gē "earth" (see Gaia) + stem of daiein "to divide," from PIE.

Is a manifold closed?

In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only non-compact components.

What is a 0 manifold?

In a manifold of dimension zero, every singleton subset is homeomorphic to R0 and so is open. Thus, a zero-dimensional manifold is a discrete set. By second countability, this discrete set must be countable.

What is manifold classification Class 11?

If a population is divided into a number of mutually exclusive classes according to some given characteristic and then each class is divided by reference to some second, third, etc. characteristic, the final grouping is called a manifold classification.

Are Riemannian manifolds connected?

is not simply-connected, so that there is a covering map. This is an immersion, and so the universal cover of any Riemannian manifold automatically inherits a Riemannian metric. More generally, but by the same principle, any covering space of a Riemannian manifold inherits a Riemannian metric.

What is Riemannian geometry used for?

Riemannian Geometry studies smooth manifolds using a Riemannian metric. Locally, manifolds have properties of Euclidean spaces or other topological spaces, often in higher dimensions. Riemannian metrics express distances by means of smooth positive definite bilinear forms.

Is Riemannian geometry non Euclidean?

Riemannian geometry, also called elliptic geometry, one of the non-Euclidean geometries that completely rejects the validity of Euclid's fifth postulate and modifies his second postulate.

What is geodesic in differential geometry?

“Definition” 7.1. 1. Let S be a surface. A curve α : I → S parametrized by arc length is called a geodesic if for any two points P = α(s1),Q = α(s2) on the curve which are sufficiently close to each other, the piece of the trace of α between P and Q is the shortest of all curves in S which join P and Q.

Is an orbit a geodesic?

Whatever the origin of the velocity: every orbit is a geodesic. The criterium is that the object in motion isn't pulling G's. For any object the shape of the geodesic that it is moving along follows from both the distribution of gravity and the velocity of the object with respect to that distribution of gravity.

What is Geodetic Science?

Geodesy is the science of accurately measuring and understanding the Earth's geometric shape, orientation in space, and gravity field.

Why is geodesic important?

The geodesic design is a perfect marriage of the sturdy arch and the rigid triangle, which enables dome homes to be extremely strong. They can withstand high winds, earthquakes and heavy snow, making them ideal structures for any type of environment, especially in an increasingly volatile climate.

Is geodesic unique?

For every p 2 M and every v 2 TpM, there is a unique geodesic, denoted v, such that (0) = p, 0(0) = v, and the domain of is the largest possible, that is, cannot be extended.