Conjugate points

Conjugate points

In differential geometry, conjugate points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian.

Definition

Suppose p and q are points on a Riemannian manifold, and γ is a geodesic that connects p and q. Then p and q are conjugate points along γ if there exists a non-zero Jacobi field along γ that vanishes at p and q.

Recall that any Jacobi field can be written as the derivative of a geodesic variation (see the article on Jacobi fields). Therefore, if p and q are conjugate along γ, one can construct a family of geodesics which start at p and almost end at q. In particular, if γs(t) is the family of geodesics whose derivative in s at s = 0 generates the Jacobi field J, then the end point of the variation, namely γs(1), is the point q only up to first order in s. Therefore, if two points are conjugate, it is not necessary that there exist two distinct geodesics joining them.

Examples

  • On the sphere S2, antipodal points are conjugate.
  • On \mathbb{R}^n, there are no conjugate points.
  • On Riemannian manifolds with non-positive sectional curvature, there are no conjugate points.

See also


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  • conjugate points — jungtiniai taškai statusas T sritis fizika atitikmenys: angl. conjugate points vok. konjugierte Punkte, m rus. сопряжённые точки, f pranc. points conjugués, m …   Fizikos terminų žodynas

  • conjugate — conjugable /kon jeuh geuh beuhl/, adj. conjugably, adv. conjugative, adj. conjugator, n. v. /kon jeuh gayt /; adj., n. /kon jeuh git, gayt /, v., conjugated, conjugating, adj., n. v.t. 1. Gram. a. to inflect (a verb) …   Universalium

  • conjugate — con•ju•gate v. [[t]ˈkɒn dʒəˌgeɪt[/t]] adj., n. [[t]ˈkɒn dʒə gɪt, ˌgeɪt[/t]] v. gat•ed, gat•ing, adj. n. 1) gram. a) to recite or display all or some subsets of the inflected forms of (a verb) in a fixed order: to conjugate the present tense of… …   From formal English to slang

  • points conjugués — jungtiniai taškai statusas T sritis fizika atitikmenys: angl. conjugate points vok. konjugierte Punkte, m rus. сопряжённые точки, f pranc. points conjugués, m …   Fizikos terminų žodynas

  • conjugate foci — noun (physics) Two points such that rays of light diverging from either of them are brought to a focus at the other • • • Main Entry: ↑conjugate conjugate foci (optics) Two points such that each is the focus for rays proceeding from the other • • …   Useful english dictionary

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  • Conjugate diameters — Conjugate Con ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See {Join}.] 1. United in pairs; yoked together; coupled. [1913 Webster] 2. (Bot.) In single pairs;… …   The Collaborative International Dictionary of English

  • Conjugate focus — Conjugate Con ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See {Join}.] 1. United in pairs; yoked together; coupled. [1913 Webster] 2. (Bot.) In single pairs;… …   The Collaborative International Dictionary of English

  • Conjugate mirrors — Conjugate Con ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See {Join}.] 1. United in pairs; yoked together; coupled. [1913 Webster] 2. (Bot.) In single pairs;… …   The Collaborative International Dictionary of English

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