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Jacobi field

In , a Jacobi field along a \gamma on a is a J that satisfies the \nabla_{\dot{\gamma}} \nabla_{\dot{\gamma}} J + R(J, \dot{\gamma}) \dot{\gamma} = 0, where \nabla denotes the and R is the . This captures the infinitesimal behavior of nearby geodesics, providing a to how geodesics deviate from one another under small perturbations. Jacobi fields arise naturally as variation fields in geodesic variations, where a family of geodesics \Gamma(s, t) = \gamma_s(t) parameterized by s has the property that the transverse derivative V = \frac{\partial}{\partial s} \big|_{s=0} \Gamma satisfies the Jacobi equation along \gamma = \gamma_0. Conversely, every Jacobi field is the variation field of some geodesic variation, ensuring a bijective correspondence between such fields and deformations of the . For initial conditions J(0) and \nabla_{\dot{\gamma}}(0) J specified at the starting point, there exists a unique Jacobi field along \gamma, and the space of all Jacobi fields has $2n on an n-dimensional manifold, reflecting the freedom in initial and . A key application of Jacobi fields is in identifying conjugate points along a , where a non-trivial Jacobi field vanishes at both endpoints, signaling the failure of the geodesic to be locally length-minimizing beyond that point. This concept is central to theorems on the injectivity radius, , and comparison geometry in Riemannian manifolds, such as estimates on the diameter and sphere theorems that classify manifolds with positive curvature. Jacobi fields also play a role in the second variation of or energy functionals, quantifying stability and oscillations of geodesics influenced by .

Fundamentals

Definition

In , geodesics are curves that locally extremize length, serving as the analogs of straight lines in curved spaces. Jacobi fields arise as solutions to the variational equation associated with geodesics, capturing the infinitesimal behavior of nearby geodesics. Specifically, they are defined through a one-parameter family of curves \Gamma: [0, L] \times (-\epsilon, \epsilon) \to M on a (M, g), where for each fixed t \in (-\epsilon, \epsilon), the curve \gamma_t(s) = \Gamma(s, t) (with s as the arc-length parameter) is a geodesic, and \gamma_0(s) = \gamma(s) is the reference geodesic from p = \gamma(0) to q = \gamma(L). The Jacobi field J along \gamma is then the variation given formally by J(s) = \left. \frac{\partial}{\partial t} \Gamma(s, t) \right|_{t=0}, which satisfies the Jacobi equation (derived in subsequent sections). The space of all Jacobi fields along a given \gamma on an n-dimensional manifold forms a real of dimension $2n, encompassing trivial Jacobi fields generated by reparameterizations of \gamma and along it, as well as non-trivial fields reflecting the manifold's . These fields are closely tied to the spaces at the endpoints, as any pair of vectors \xi \in T_p M and \eta \in T_q M determines initial and final conditions J(0) = \xi and J(L) = \eta for a unique Jacobi field along \gamma, assuming no conjugate points along the segment.

Properties

Jacobi fields along a in a form a of $2n, where n is the of the manifold, as they satisfy a linear second-order and thus admit linear combinations as solutions. Specifically, if J_1 and J_2 are Jacobi fields along the same \gamma, then for any scalars a, b \in \mathbb{R}, the field a J_1 + b J_2 is also a Jacobi field, reflecting the inherent in their variational origin. Among Jacobi fields, trivial ones arise from variations that merely reparameterize the or involve parallel translation along it, such as J(t) = \dot{\gamma}(t) or J(t) = t \dot{\gamma}(t), and these exhibit no dependence on the manifold's . These trivial fields span a two-dimensional and correspond to changes in the 's speed or direction without altering its path essentially. Non-trivial Jacobi fields encode the effects of the manifold's geometry through dependence on the , capturing how sectional curvatures influence the evolution of nearby along \gamma. In particular, the behavior of these fields reflects local curvature variations, distinguishing manifolds with positive, negative, or zero curvature in their geodesic spreading properties. The for Jacobi fields states that, given initial conditions J(0) = X and \nabla_{\dot{\gamma}} J(0) = Y for vectors [X, Y](/page/X&Y) \in T_{\gamma(0)}M, there exists a unique Jacobi field J along \gamma satisfying these conditions, guaranteed by the standard existence and uniqueness results for linear ODEs on manifolds. Jacobi fields admit an orthogonal decomposition into components parallel and perpendicular to the direction \dot{\gamma}, where the perpendicular fields, satisfying g(J(t), \dot{\gamma}(t)) = 0 for all t, evolve according to a simplified structure independent of tangential influences and span a of dimension $2(n-1). This decomposition facilitates analysis of transverse behavior, isolating curvature effects in the normal bundle.

Motivation and Derivation

Geodesic Variations

A geodesic variation provides a framework for studying the local behavior of by considering a smooth one-parameter family of curves on a . Specifically, it is defined as a smooth map H: (-\epsilon, \epsilon) \times [a, b] \to M from an open interval around zero in the parameter space to the manifold M, such that H(0, t) = \gamma(t) traces the base \gamma, and each curve \gamma_s(t) = H(s, t) for fixed s is itself a . This setup allows for the analysis of perturbations around \gamma, where the variation field V(t) = \frac{\partial}{\partial s} H(s, t) \big|_{s=0} along \gamma captures the tangential displacement induced by nearby curves in the family. The first variation of the or energy functional associated with such a family vanishes precisely when the base curve is a , confirming that geodesics are critical points in the space of curves. This vanishing condition, \delta E(\gamma) = 0 or equivalently \delta L(\gamma) = 0, arises from integrating the inner product of the of the variation field with the along \gamma, yielding zero for proper variations with fixed endpoints. In essence, it characterizes the stationarity of the geodesic under small deformations, providing the foundational link between variational principles and the geodesic equation. Geodesic variations thus measure the "infinitesimal rigidity" of a by quantifying how nearby curves deviate from it, with the variation field describing the rate of this separation or convergence. This deviation reflects the intrinsic of the manifold, offering insight into the and of geodesics without requiring explicit solutions to governing equations. Jacobi fields, as the variation fields of such families, embody this notion and form the tangent space to the set of geodesics at \gamma. The concept originated in Carl Gustav Jacob Jacobi's investigations into variations of paths in during the 1840s, where he applied to planetary motion problems, introducing ideas akin to conjugate points for extremal paths. These methods were later extended to by Gaston Darboux and others in the late , particularly through analyses of geodesics on surfaces like ellipsoids, bridging and intrinsic geometric properties.

Jacobi Equation

The Jacobi equation arises from the study of geodesic variations in a . Consider a smooth one-parameter family of curves \gamma(s, t) on a M, where s is the variation parameter and t parameterizes each curve, such that \gamma(0, t) = \gamma(t) is a satisfying the \nabla_{\partial/\partial t} \partial \gamma / \partial t = 0. The Jacobi field J along \gamma is defined as the variation field J(t) = \partial \gamma / \partial s \big|_{s=0}. To derive the governing equation, differentiate the geodesic equation with respect to the variation parameter s and evaluate at s=0. The geodesic condition implies \nabla_{\partial/\partial t} \partial \gamma / \partial t = 0 for all s. Assuming the coordinate vector fields commute, [\partial / \partial s, \partial / \partial t] = 0, the torsion-freeness of the gives \nabla_{\partial/\partial t} \partial / \partial s = \nabla_{\partial / \partial s} \partial / \partial t. Differentiating the geodesic equation with respect to s yields \nabla_{\partial / \partial s} \left( \nabla_{\partial/\partial t} \frac{\partial \gamma}{\partial t} \right) = 0. Using the definition of the , R\left( \frac{\partial}{\partial s}, \frac{\partial}{\partial t} \right) \frac{\partial \gamma}{\partial t} = \nabla_{\partial / \partial s} \nabla_{\partial / \partial t} \frac{\partial \gamma}{\partial t} - \nabla_{\partial / \partial t} \nabla_{\partial / \partial s} \frac{\partial \gamma}{\partial t} - \nabla_{[\partial / \partial s, \partial / \partial t]} \frac{\partial \gamma}{\partial t}, and substituting the differentiated geodesic equation and commutativity, this simplifies at s=0 with \partial \gamma / \partial t = \dot{\gamma} and \partial \gamma / \partial s = J to \nabla_{\dot{\gamma}} \nabla_{\dot{\gamma}} J + R(J, \dot{\gamma}) \dot{\gamma} = 0. This is the explicit Jacobi equation, often written in terms of the covariant derivative D/dt along \gamma: \frac{D^2 J}{dt^2} + R(J, \dot{\gamma}) \dot{\gamma} = 0, where R is the Riemann curvature tensor. The equation is a second-order linear ordinary differential equation (ODE) for vector fields J along the geodesic \gamma, with the curvature term R(J, \dot{\gamma}) \dot{\gamma} acting as a "potential" that encodes the manifold's geometry. For variations with fixed endpoints, such as those where \gamma(s, 0) = \gamma(0) and \gamma(s, L) = \gamma(L) for some length L, the Jacobi field satisfies the boundary conditions J(0) = 0 and J(L) = 0. These conditions arise because the variation field vanishes at the fixed points.

Solutions and Geometric Interpretation

Solution Methods

The Jacobi equation along a geodesic \gamma: [0, L] \to M in a (M, g) is a linear second-order system of ordinary differential equations (ODEs) for the components of the Jacobi field J(t), given by \nabla^2_{\dot{\gamma}} J + R(J, \dot{\gamma})\dot{\gamma} = 0, where \nabla is the and R is the . Solutions exist and are unique for specified initial conditions J(0) and \nabla_{\dot{\gamma}} J(0), as guaranteed by the Picard-Lindelöf theorem applied to the equivalent first-order system, due to the smoothness of the and curvature. In Fermi coordinates (t, x^1, \dots, x^{n-1}) adapted to \gamma, where \gamma(t) = (t, 0, \dots, 0) and the metric satisfies g_{tt} = 1, g_{ti} = 0, and g_{ij} = \delta_{ij} + O(|x|^2), the Jacobi equation for transverse Jacobi fields (perpendicular to \dot{\gamma}) reduces to the matrix ODE \frac{d^2}{dt^2} \mathbf{J} + K(t) \mathbf{J} = 0, where \mathbf{J} = (J^1, \dots, J^{n-1})^T and K(t) is the (n-1) \times (n-1) symmetric matrix with entries K^i_j(t) = R^i{}_{t j t}(\gamma(t)), encoding the curvature operator restricted to the normal bundle. This coordinate reduction simplifies numerical or analytical treatment, as the Christoffel symbols vanish along \gamma, isolating the curvature's influence on geodesic deviation. To solve explicitly, one may decompose J(t) in a parallel orthonormal frame \{e_1(t), \dots, e_{n-1}(t)\} along \gamma orthogonal to \dot{\gamma}(t), obtained via of an initial at \gamma(0). Writing J(t) = \sum_{i=1}^{n-1} f^i(t) e_i(t), the Jacobi equation decouples into the \ddot{f}^i(t) + \sum_{j=1}^{n-1} \kappa^i_j(t) f^j(t) = 0 for i = 1, \dots, n-1, where \kappa^i_j(t) = g(R(e_i(t), \dot{\gamma}(t))\dot{\gamma}(t), e_j(t)) are the components of the operator. In cases where the frame aligns with principal directions of (e.g., when s dominate in specific planes), the system may further simplify to uncoupled scalar s \ddot{f}^i(t) + \kappa_i(t) f^i(t) = 0, with \kappa_i(t) the of the plane spanned by \dot{\gamma}(t) and e_i(t). On complete Riemannian manifolds, the Hopf-Rinow theorem ensures geodesics are defined globally, so initial value problems for the Jacobi equation admit solutions on (-\infty, \infty), with asymptotic behavior governed by the sign of the sectional curvatures: positive curvature leads to oscillatory solutions, while negative curvature yields exponential growth or decay. For boundary value problems relevant to conjugate points or stability, the Jacobi equation forms a self-adjoint Sturm-Liouville system on finite intervals, whose eigenvalues determine the Morse index of the geodesic via oscillation theorems, analogous to classical Sturm oscillation theory adapted to variable coefficients from curvature.

Conjugate Points

In a M, consider a \gamma: [0, L] \to M parameterized by with \gamma(0) = p and \gamma(L) = q. The point q is said to be conjugate to p along \gamma if there exists a non-trivial Jacobi field J along \gamma satisfying J(0) = 0 and J(L) = 0. This condition implies that the differential of the \exp_p at L \gamma'(0) is degenerate, as the kernel of d(\exp_p) consists precisely of such Jacobi fields vanishing at the endpoints. Geometrically, conjugate points mark locations where nearby geodesics emanating from p begin to reconverge, signaling the potential multiplicity of paths connecting p to q. Conjugate points arise as the zeros of non-trivial solutions to the Jacobi equation along \gamma that satisfy the initial vanishing condition J(0) = 0. These zeros indicate the points where the linear approximation of variations fails to be injective, corresponding to the first instances of linear dependence among the images of nearby initial velocities under the . Along a segment without conjugate points, the map \exp_p restricted to the corresponding ball in T_p M is a onto its image, ensuring that \gamma remains the unique connecting p to points before the first such zero. The absence of conjugate points along a segment \gamma|_{[0,t]} implies that this segment is locally length-minimizing between its endpoints, as any nearby variation would have non-negative second variation by the index form's positivity. More strongly, up to the first conjugate point, \gamma is globally distance-minimizing within the injectivity domain of \exp_p, beyond which shorter paths may exist due to the . This property underscores the geometric role of conjugate points as boundaries of the region where geodesics uniquely realize distances. The relates the positions of conjugate points to bounds by comparing Jacobi fields in M to those in model spaces of constant . Specifically, if the s of M are bounded above by a constant \kappa, then the norm |J(t)| of a Jacobi field J with J(0) = 0 satisfies a differential inequality |J|'' + \kappa |J| \geq 0, implying that the first zero of J occurs no earlier than the corresponding conjugate point distance \pi / \sqrt{\kappa} in the sphere of \kappa. This comparison highlights how positive curvatures accelerate the appearance of conjugate points, while non-positive curvatures preclude them entirely along any .

Applications

Second Variation Calculus

The second variation of the energy functional along a geodesic provides a quadratic form that measures the stability of the geodesic as a critical point of the variational problem. For a geodesic \gamma: [a, b] \to M on a Riemannian manifold M, the energy functional is defined as E(\gamma) = \frac{1}{2} \int_a^b \|\gamma'(t)\|^2 \, dt. The second variation \delta^2 E at \gamma for an infinitesimal variation field J along \gamma with J(a) = J(b) = 0 is given by \delta^2 E(J) = \int_a^b \left\langle \frac{DJ}{dt}, \frac{DJ}{dt} \right\rangle - \left\langle R(J, \gamma') \gamma', J \right\rangle \, dt, where R is the Riemann curvature tensor and \frac{D}{dt} denotes the covariant derivative along \gamma. This expression defines the index form I(J, J) = \delta^2 E(J), a symmetric bilinear form on the space of vector fields along \gamma vanishing at the endpoints. Jacobi fields play a central role as the critical points of this variational problem. Specifically, for variations generating Jacobi fields J satisfying the Jacobi equation \frac{D^2 J}{dt^2} + R(J, \gamma') \gamma' = 0 with fixed endpoints, the first variation vanishes, and the second variation simplifies such that I(J, J) = 0. For a minimizing , the index form is , meaning I(J, J) \geq 0 for all admissible J, with equality precisely when J is a Jacobi field. This positivity condition characterizes local minimizers of the energy functional among nearby curves. The Morse index of the geodesic segment \gamma is defined as the number of negative eigenvalues of the index form I, counting multiplicities, which quantifies the number of directions in which the energy decreases. By the Morse index theorem for geodesics, this index equals the sum of the multiplicities of the conjugate points along \gamma between a and b. Conjugate points thus mark points where the index form loses , signaling the onset of instability. In the context of applied to the space of s, the index form connects to global topological properties. The Bott-Samelson theorem provides a method to compute the Morse index for geodesics in symmetric spaces by decomposing the loop space into cells via a combinatorial construction involving the , enabling explicit determination of Betti numbers and torsion-freeness. This links the local geometry captured by Jacobi fields to the type of the path space.

Geodesic Stability

Jacobi fields play a central role in assessing the of within . A segment from point p to q is considered if it minimizes the functional locally among nearby curves connecting the same endpoints. This holds if and only if there are no conjugate points along the segment, in which case the index form is positive definite on the space of admissible variation fields vanishing at the endpoints. The absence of conjugate points ensures that the is a , preventing the existence of non-trivial Jacobi fields that vanish at two distinct points along the , which would otherwise indicate a loss of minimality. In , Jacobi fields provide the mathematical framework for the , which quantifies the relative acceleration between nearby due to . This models tidal forces acting on extended objects in free fall, such as satellites or test particles in a . Specifically, for a deviation vector \xi transverse to the tangent vector u of a , the takes the form \frac{D^2 \xi}{dt^2} = -R(\xi, u) u, where R denotes the , and solutions to this linear differential are precisely the Jacobi fields along the . This relation arises from the linearized for a family of nearby curves and directly links the geometry's to physical effects like the focusing of light rays or the stretching of in strong fields. Beyond the conjugate locus, Jacobi fields facilitate the construction of the cut locus of a point p on a manifold, which consists of the first points along from p where minimality is lost. The cut locus is determined as the set of points q where either q is the first conjugate point along a minimizing from p, or multiple distinct minimizing from p meet at q. At such points, non-trivial Jacobi fields indicate the boundary beyond which the geodesic ceases to be length-minimizing, marking the transition to regions where shorter paths exist via alternative routes. This construction is essential for understanding the global topology and injectivity radius of the manifold. In advanced applications, such as the Hawking-Penrose singularity theorems, conjugate points along —detected via vanishing Jacobi fields—signal the incompleteness of . These theorems establish that under conditions like the presence of trapped surfaces and non-positive average , must terminate at conjugate points in finite affine parameter, implying the existence of where becomes unbounded. The role of Jacobi fields here underscores how local conditions propagate to global geodesic incompleteness, providing a rigorous basis for predicting and the origin in cosmological models.

Examples

Constant Curvature Manifolds

In manifolds of constant K, the Jacobi equation simplifies significantly for the perpendicular component of Jacobi fields along a unit-speed \gamma(t). Specifically, if J(t) is a Jacobi field perpendicular to \gamma'(t), it can be expressed as J(t) = u(t) E(t), where E(t) is the parallel transport along \gamma of an initial perpendicular vector E(0) \in T_{\gamma(0)}M with g(E(0), \gamma'(0)) = 0, and u(t) satisfies the scalar u''(t) + K u(t) = 0. This decoupling arises because the operator acts as multiplication by K on perpendicular vectors in such spaces. For the sphere S^n with constant sectional curvature K = 1 (unit sphere), the general solution for a perpendicular Jacobi field is J(t) = \cos(t) \, \mathrm{PT}_t(J(0)^\perp) + \sin(t) \, \mathrm{PT}_t(J'(0)^\perp), where \mathrm{PT}_t denotes parallel transport along \gamma from t=0 to t, and J(0)^\perp, J'(0)^\perp are the perpendicular components of the initial value and derivative, respectively. This form captures both the "radial" contribution from the initial displacement perpendicular to the geodesic and the "transverse" contribution from the initial variation rate. Equivalently, in terms of initial data with E(0) perpendicular to \gamma'(0), it aligns with J(t) = \cos(t) \, E(t) + \sin(t) \, F(t), where E(t) is the parallel transport of the perpendicular component of J(0), and F(t) is the parallel transport of the perpendicular part of J'(0), emphasizing the oscillatory behavior due to positive curvature. In hyperbolic space H^n with constant sectional curvature K = -1, the perpendicular Jacobi field takes the form J(t) = \cosh(t) \, \mathrm{PT}_t(J(0)^\perp) + \sinh(t) \, \mathrm{PT}_t(J'(0)^\perp). This hyperbolic solution reflects the exponential divergence of nearby geodesics, with the \cosh(t) term dominating for large t from the initial displacement and the \sinh(t) term from the initial variation. The full vector expression separates the perpendicular components clearly, as the parallel transport preserves the orthogonality to \gamma'(t). Conjugate points along \gamma occur where non-trivial perpendicular Jacobi fields vanish, signaling the failure of the exponential map to be immersive. On the unit sphere (K=1), the first conjugate point is at t = \pi, where \sin(\pi) = 0, corresponding to antipodal points; beyond this distance, the exponential map develops singularities. In hyperbolic space (K=-1), the functions \sinh(t) and \cosh(t) have no positive zeros, so there are no conjugate points, and geodesics spread without focal points. In positive constant curvature spaces like , non-trivial perpendicular Jacobi fields vanishing at two distinct points along a geodesic segment of length less than \pi do not exist; only the trivial zero field satisfies this up to the first conjugate distance, ensuring local uniqueness of geodesics. This property underscores the completeness of the within the injectivity radius on such manifolds.

Euclidean and Other Spaces

In Euclidean space \mathbb{R}^n equipped with the flat metric, the curvature tensor vanishes identically, simplifying the Jacobi equation along any geodesic \gamma(t) to \frac{D^2 J}{dt^2} = 0, where J(t) is a Jacobi field and \frac{D}{dt} denotes the covariant derivative along \gamma. The general solution takes the affine form J(t) = J(0) + t \frac{DJ}{dt}(0), reflecting linear growth without oscillation or decay. Consequently, non-trivial Jacobi fields along geodesics in \mathbb{R}^n never vanish for t > 0, implying the absence of conjugate points and ensuring that the exponential map remains a local diffeomorphism everywhere. On Riemannian manifolds admitting Killing vector fields—vector fields that generate isometries and preserve the metric—such fields restricted to a yield Jacobi fields. Specifically, if \xi is a , then J(t) = \xi(\gamma(t)) satisfies the Jacobi equation \frac{D^2 J}{dt^2} + R(J, \dot{\gamma}) \dot{\gamma} = 0, where R is the curvature operator, due to the compatibility of \xi with the . This construction is particularly relevant for spaces with symmetries, such as rotations on the round sphere or translations in , where these Jacobi fields describe infinitesimal deformations under group actions without altering lengths. For manifolds with variable curvature, such as surfaces of revolution like the , Jacobi fields can be computed via in adapted coordinates, leveraging conserved quantities analogous to the Clairaut relation for geodesics. On a of revolution parametrized by meridional distance s and azimuthal angle \theta, the Jacobi equation decouples into ordinary differential equations, with solutions involving elliptic functions that account for the K(s) varying between positive and negative values. These fields exhibit oscillatory behavior near regions of positive curvature (e.g., the outer ) and exponential divergence in negative curvature zones (e.g., the inner ), leading to conjugate points where nearby geodesics intersect. A non-trivial illustration arises on the triaxial , a with variable K > 0 that decreases from poles to . Jacobi fields along geodesics, solved using as pioneered by Jacobi, vanish at focal (conjugate) points, where the becomes singular and nearby geodesics converge. This manifests in four-cusped conjugate loci for generic starting points, highlighting focal points as sites of geodesic instability. In practical contexts, such as modeling Earth's oblate spheroidal surface for , these Jacobi fields quantify variations in great-ellipse paths, with conjugate points indicating regions where navigational focal loci emerge due to the mild ellipticity.