Jacobi equation
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In
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi.


Definitions and properties

Jacobi fields can be obtained in the following way: Take a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
one parameter family of geodesics \gamma_\tau with \gamma_0=\gamma, then :J(t)=\left.\frac\_ is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic \gamma. A vector field ''J'' along a geodesic \gamma is said to be a Jacobi field if it satisfies the Jacobi equation: :\fracJ(t)+R(J(t),\dot\gamma(t))\dot\gamma(t)=0, where ''D'' denotes the covariant derivative with respect to the Levi-Civita connection, ''R'' the
Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
, \dot\gamma(t)=d\gamma(t)/dt 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 \gamma_\tau describing the field (as in the preceding paragraph). The Jacobi equation is a
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
, second order
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
; in particular, values of J and \fracJ at one point of \gamma uniquely determine the Jacobi field. Furthermore, the set of Jacobi fields along a given geodesic forms a real
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of dimension twice the dimension of the manifold. As trivial examples of Jacobi fields one can consider \dot\gamma(t) and t\dot\gamma(t). These correspond respectively to the following families of reparametrisations: \gamma_\tau(t)=\gamma(\tau+t) and \gamma_\tau(t)=\gamma((1+\tau)t). Any Jacobi field J can be represented in a unique way as a sum T+I, where T=a\dot\gamma(t)+bt\dot\gamma(t) is a linear combination of trivial Jacobi fields and I(t) is orthogonal to \dot\gamma(t), for all t. The field I then corresponds to the same variation of geodesics as J, only with changed parameterizations.


Motivating example

On a
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...
, the geodesics through the North pole are great circles. Consider two such geodesics \gamma_0 and \gamma_\tau with natural parameter, t\in ,\pi/math>, separated by an angle \tau. The geodesic distance :d(\gamma_0(t),\gamma_\tau(t)) \, is :d(\gamma_0(t),\gamma_\tau(t))=\sin^\bigg(\sin t\sin\tau\sqrt\bigg). Computing this requires knowing the geodesics. The most interesting information is just that :d(\gamma_0(\pi),\gamma_\tau(\pi))=0 \,, for any \tau. Instead, we can consider the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
with respect to \tau at \tau=0: :\frac\bigg, _d(\gamma_0(t),\gamma_\tau(t))=, J(t), =\sin t. Notice that we still detect the intersection of the geodesics at t=\pi. Notice further that to calculate this derivative we do not actually need to know :d(\gamma_0(t),\gamma_\tau(t)) \,, rather, all we need do is solve the equation :y''+y=0 \,, for some given initial data. Jacobi fields give a natural generalization of this phenomenon to arbitrary Riemannian manifolds.


Solving the Jacobi equation

Let e_1(0)=\dot\gamma(0)/, \dot\gamma(0), and complete this to get an
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
basis \big\ at T_M.
Parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
it to get a basis \ all along \gamma. This gives an orthonormal basis with e_1(t)=\dot\gamma(t)/, \dot\gamma(t), . The Jacobi field can be written in co-ordinates in terms of this basis as J(t)=y^k(t)e_k(t) and thus :\fracJ=\sum_k\frace_k(t),\quad\fracJ=\sum_k\frace_k(t), and the Jacobi equation can be rewritten as a system :\frac+, \dot\gamma, ^2\sum_j y^j(t)\langle R(e_j(t),e_1(t))e_1(t),e_k(t)\rangle=0 for each k. This way we get a linear ordinary differential equation (ODE). Since this ODE has
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
coefficients we have that solutions exist for all t and are unique, given y^k(0) and '(0), for all k.


Examples

Consider a geodesic \gamma(t) with parallel orthonormal frame e_i(t), e_1(t)=\dot\gamma(t)/, \dot\gamma, , constructed as above. * The vector fields along \gamma given by \dot \gamma(t) and t\dot \gamma(t) are Jacobi fields. * In Euclidean space (as well as for spaces of constant zero
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...
) Jacobi fields are simply those fields linear in t. *For Riemannian manifolds of constant negative sectional curvature -k^2, any Jacobi field is a linear combination of \dot\gamma(t), t\dot\gamma(t) and \exp(\pm kt)e_i(t), where i>1. *For Riemannian manifolds of constant positive sectional curvature k^2, any Jacobi field is a linear combination of \dot\gamma(t), t\dot\gamma(t), \sin(kt)e_i(t) and \cos(kt)e_i(t), where i>1. *The restriction of a Killing vector field to a geodesic is a Jacobi field in any Riemannian manifold.


See also

* Conjugate points *
Geodesic deviation equation In general relativity, if two objects are set in motion along two initially parallel trajectories, the presence of a tidal gravitational force will cause the trajectories to bend towards or away from each other, producing a relative acceleration be ...
* Rauch comparison theorem * N-Jacobi field


References

* Manfredo Perdigão do Carmo. Riemannian geometry. Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992. xiv+300 pp. * Jeff Cheeger and David G. Ebin. Comparison theorems in Riemannian geometry. Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. x+168 pp. *
Shoshichi Kobayashi was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi. His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie alg ...
and
Katsumi Nomizu was a Japanese-American mathematician known for his work in differential geometry. Life and career Nomizu was born in Osaka, Japan on the first day of December, 1924. He studied mathematics at Osaka University, graduating in 1947 with a Maste ...
. Foundations of differential geometry. Vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. xvi+468 pp. * Barrett O'Neill. Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc. arcourt Brace Jovanovich, Publishers New York, 1983. xiii+468 pp. {{ISBN, 0-12-526740-1 Riemannian geometry Equations