In Riemannian geometry, a Jacobi field is a vector field along a geodesic γ {\displaystyle \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 one parameter family of geodesics γ τ {\displaystyle \gamma _{\tau }} with γ 0 = γ {\displaystyle \gamma _{0}=\gamma } , then
J ( t ) = ∂ γ τ ( t ) ∂ τ | τ = 0 {\displaystyle J(t)=\left.{\frac {\partial \gamma _{\tau }(t)}{\partial \tau }}\right|_{\tau =0}}
is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic γ {\displaystyle \gamma } . A vector field J along a geodesic γ {\displaystyle \gamma } is said to be a Jacobi field if it satisfies the Jacobi equation:
D 2 d t 2 J ( t ) + R ( J ( t ) , γ ˙ ( t ) ) γ ˙ ( t ) = 0 , {\displaystyle {\frac {D^{2}}{dt^{2}}}J(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, γ ˙ ( t ) = d γ ( t ) / d t {\displaystyle {\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 γ τ {\displaystyle \gamma _{\tau }} describing the field (as in the preceding paragraph). The Jacobi equation is a linear, second order ordinary differential equation; in particular, values of J {\displaystyle J} and D d t J {\displaystyle {\frac {D}{dt}}J} at one point of γ {\displaystyle \gamma } uniquely determine the Jacobi field. Furthermore, the set of Jacobi fields along a given geodesic forms a real vector space of dimension twice the dimension of the manifold. As trivial examples of Jacobi fields one can consider γ ˙ ( t ) {\displaystyle {\dot {\gamma }}(t)} and t γ ˙ ( t ) {\displaystyle t{\dot {\gamma }}(t)} . These correspond respectively to the following families of reparametrizations: γ τ ( t ) = γ ( τ + t ) {\displaystyle \gamma _{\tau }(t)=\gamma (\tau +t)} and γ τ ( t ) = γ ( ( 1 + τ ) t ) {\displaystyle \gamma _{\tau }(t)=\gamma ((1+\tau )t)} . Any Jacobi field J {\displaystyle J} can be represented in a unique way as a sum T + I {\displaystyle T+I} , where T = a γ ˙ ( t ) + b t γ ˙ ( t ) {\displaystyle T=a{\dot {\gamma }}(t)+bt{\dot {\gamma }}(t)} is a linear combination of trivial Jacobi fields and I ( t ) {\displaystyle I(t)} is orthogonal to γ ˙ ( t ) {\displaystyle {\dot {\gamma }}(t)} , for all t {\displaystyle t} . The field I {\displaystyle I} then corresponds to the same variation of geodesics as J {\displaystyle J} , only with changed parametrizations.
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