First Variation Formula

Let \((M,g)\) be a smooth Riemannian manifold. Write \(\nabla\) for its Levi–Civita connection and \(T_pM\) for the tangent space at \(p\). The coordinate conventions are those in tangent spaces and derivations.

A variation replaces one curve by a nearby family of curves. Differentiating the energy of that family tells us how the energy responds to an infinitesimal displacement. The first variation formula separates this response into an interior term, involving acceleration, and endpoint terms.

This separation explains why fixed-endpoint critical curves satisfy the geodesic equation. We first describe variations and their vector fields, then derive the formula by differentiating under the integral and integrating by parts.

Definition 1. A variation of a curve \(c:[a,b]\to M\) is a continuous map \(\Gamma:(-\varepsilon,\varepsilon)\times[a,b]\to M\), smooth on the pieces described below, with \(\Gamma(0,t)=c(t)\). It is proper when \(\Gamma(s,a)=c(a)\) and \(\Gamma(s,b)=c(b)\) for every \(s\). It is a variation through geodesics when \(t\mapsto\Gamma(s,t)\) is a geodesic for each fixed \(s\).

Remark 2. Definition 1 requires differentiability of \(c(s,t)\) for each \(t \in [a,b]\). Formally, we require that there exists a subdivision of \([a,b]\), denoted \(a = t_0 < t_1 < \cdots < t_{k+1} = b\) such that the restriction of \(c(s,t)\) to each \((-\varepsilon,\varepsilon) \times [t_i, t_{i+1}]\) for each \(i = 0,1,\cdots,k\) is differentiable. Some textbooks, such as (Carmo 1992), follow this convention. The consequence of this formality is an additional term in the variation of the energy functional.

Remark 3. Especially for studying the Jacobi fields and related topics (in Section Jacobi Fields and Conjugate Points, we often want to fix a point in “time”, and examine the family of curves induced by the variation of some curve \(c\). We customarily write \(\Gamma_t (s)\), for some fixed \(t\), and call this a transversal curve of the variation \(\Gamma\). We can examine the velocity vector of a transversal curve at \(s=0\), i.e. defined by \(V(t) = \pd{\Gamma}{s}(s,t)|_{s=0}\), and this is a (piecewise differentiable) vector field along curve \(c\). \(V(t)\) is called the variational field of \(\Gamma\).

To use vector fields as test directions, we need to realize each one as the derivative of an actual variation. The exponential map provides such a variation by moving each point of the curve a short distance in the prescribed direction.

Proposition 4. Given a continuous, piecewise smooth field \(V(t)\), along the piecewise differentiable curve \(c:[0,a] \to M\), there exists a variation \(\Gamma : (-\varepsilon,\varepsilon) \times [0,a] \to M : (s,t) \mapsto \Gamma(s,t)\) such that \(V(t)\) is the variational field of \(\Gamma\). In addition, if \(V(0) = V(a) = 0\), then it is possible to choose \(\Gamma\) as a proper variation.

The common parameter interval can be seen directly by considering \(F(s,t)=(c(t),sV(t))\). The inverse image of the exponential domain is an open subset of \(\mathbb R\times[0,a]\) containing the compact set \(\{0\}\times[0,a]\). A finite collection of product neighbourhoods therefore contains a strip \((-\varepsilon,\varepsilon)\times[0,a]\). Taking a common finite subdivision for \(c\) and \(V\) ensures that the constructed variation is smooth on each required closed piece.

Proof. Assume the piecewise smooth field \(V\) is continuous on \([0,a]\), as is usual for a variational field. The image of \(t\mapsto(c(t),0)\) is a compact subset of the zero section of \(TM\). The domain of the smooth exponential map is an open neighbourhood of that zero section. Compactness and continuity therefore give \(\varepsilon>0\) such that \((c(t),sV(t))\) lies in this domain whenever \(|s|<\varepsilon\) and \(0\le t\le a\). Set

\[\Gamma(s,t)=\exp_{c(t)}(sV(t)).\]

This map is continuous and smooth on each piece, and \(\Gamma(0,t)=c(t)\). Since \((\dif\exp_{c(t)})_0\) is the identity, \(\partial_s\Gamma(0,t)=V(t)\). If the endpoint values of \(V\) are zero, both endpoints of \(\Gamma\) are fixed. This also covers the case \(V\equiv0\). ◻

Remark 5. We can study the length and energy functionals in (3) and (3), respectively, as applied to the neighbourhood of curves in a variation \(\Gamma\) of curve \(c\). In the sequel, it is notationally more compact to write the energy functional of a variation by only indicating its dependence on the “variational parameter” \(s\), i.e. let \(L(s)\) denote the length of the variational curve \(\Gamma_s (t)\). We will express variations of both the length and energy functionals, but the energy functional is convenient for later variation formulas because its integrand is quadratic in the velocity. Again, \(E(s)\) denotes the energy functional associated with the variational curve \(\Gamma_s (t)\).

Remark 6. Take a curve \(c: [0,a] \to M\), and define \(L(c)\) and \(E(c)\), as in (3) and (3), respectively. By the Cauchy-Schwarz inequality, letting \(f\equiv1\) and \(g\equiv |\od{c}{t}|\), we have

\[\left( \int_0^a fg \dif t \right)^2 \leq \left( \int_0^a f^2 \dif t \right) \left( \int_0^a g^2 \dif t \right) ~,\]

With the convention \(E(c)=\frac12\int_0^a|\dot c|^2\,dt\), this gives \(L(c)^2\le2aE(c)\). Equality holds exactly when the speed is constant almost everywhere; for a continuous piecewise smooth curve this is the constant-speed parametrization condition. Here \(a>0\).

As a consequence of Remark 6, we have the following result.

Lemma 7. Let \(\gamma:[0,a]\to M\), \(a>0\), be a constant-speed minimizing geodesic from \(p\) to \(q\). Every continuous piecewise smooth curve \(c:[0,a]\to M\) with the same endpoints satisfies \(E(c)\ge E(\gamma)\). Equality holds precisely when \(c\) is a constant-speed minimizing geodesic from \(p\) to \(q\); it need not have the same image as \(\gamma\).

Proof. The length-energy comparison gives

\[2aE(\gamma)=d(p,q)^2\le L(c)^2\le2aE(c).\]

Equality throughout requires both \(L(c)=d(p,q)\) and constant speed. A length-minimizing curve cannot have a corner: in a convex normal neighbourhood, its short pieces must coincide with the unique minimizing radial geodesic. Thus a constant-speed minimizing piecewise smooth curve is a geodesic. Conversely, every such minimizing geodesic has energy \(d(p,q)^2/(2a)\). Distinct minimizing geodesics can attain equality, as the semicircles between antipodal points on a sphere show. ◻

Proposition 8 (Formula for the First Variation of the Energy Functional of a Curve). Let \(c:[a,b] \to M\) be a smooth regular curve on \(M\). Let \(\Gamma : (-\varepsilon,\varepsilon) \times [a,b] \to M\) denote a variation of \(c\). Then, \(L(s)\) and \(E(s)\) are differentiable with respect to \(s\), and

\[\begin{aligned} L'(0) &= \int_a^b \left( \frac{\pd{}{t} \langle \pd{\Gamma}{s}, \pd{\Gamma}{t} \rangle}{\lVert \pd{\Gamma}{t}\rVert} - \frac{\langle \pd{\Gamma}{s}, \nabla_{\partial/\partial t} \pd{\Gamma}{t} \rangle}{\lVert \pd{\Gamma}{t}\rVert} \right) \dif t \\ E'(0) &= - \int_a^b \left\langle V(t), \nabla_{\partial/\partial t} \dpd{\Gamma}{t} \right\rangle \dif t + \left\langle \dpd{\Gamma}{t} (0,b), V(b) \right\rangle - \left\langle \dpd{\Gamma}{t} (0,a), V(a) \right\rangle \end{aligned}\tag{1}\]

where \(V(t) = \pd{\Gamma}{s}(s,t)|_{s=0}\) is the variational field of \(\Gamma\), as defined in Remark 3.

Every term in the formulas for \(L\prime(0)\) and \(E\prime(0)\) is evaluated at \(s=0\).

For length, regularity is used to avoid differentiating the norm at zero. Since \(c\) is regular on a compact interval, its speed has a positive minimum; for small \(s\), smooth dependence keeps the nearby speeds uniformly positive on each piece. Differentiation under the integral is then justified by continuity on a compact parameter rectangle. The energy integrand is smooth even at zero velocity, so its first-variation formula does not need regularity.

A useful fully integrated form of the length formula is obtained by writing \(T=\dot c/|\dot c|\):

\[L'(0)=\langle V,T\rangle\big|_a^b-\int_a^b\langle V,D_tT\rangle\,dt.\]

Indeed the unintegrated derivative is \(\int\langle D_tV,T\rangle\,dt\). This formula includes the derivative of the speed inside \(D_tT\); only for constant speed can that derivative be omitted. If \(c\) has constant speed \(c_0>0\), then \(L'(0)=E'(0)/c_0\).

Proof. Metric compatibility lets us differentiate the squared speed, and torsion-freeness interchanges the two derivatives of the variation. Integration by parts then separates the acceleration term from the boundary terms. For the energy functional, this gives

\[\begin{aligned} \dod{}{s} E(s) &= \frac12 \int_a^b \dpd{}{s} \left\langle \dpd{\Gamma}{t} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle \dif t \\ &= \int_a^b \left\langle \nabla_{\partial/\partial s} \dpd{\Gamma}{t} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle \dif t \\ &= \int_a^b \left\langle \nabla_{\partial/\partial t} \dpd{\Gamma}{s} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle \dif t \\ &= \int_a^b \left( \dpd{}{t} \left\langle \dpd{\Gamma}{s} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle - \left\langle \dpd{\Gamma}{s} (s,t), \nabla_{\partial/\partial t} \dpd{\Gamma}{t} (s,t) \right\rangle \right) \dif t \\ &= \left\langle \dpd{\Gamma}{t} (s,b), \dpd{\Gamma}{s} (s,b) \right\rangle - \left\langle \dpd{\Gamma}{t} (s,a), \dpd{\Gamma}{s} (s,a) \right\rangle - \int_a^b \left\langle \dpd{\Gamma}{s}, \nabla_{\partial/\partial t} \dpd{\Gamma}{t} \right\rangle \dif t ~, \end{aligned}\]

where the second line follows from the metric property of the Levi-Civita connection, the third line follows from the torsion-free property, and the fourth and fifth lines follow from direct application of definition and direct calculation. Evaluating the last line at \(s=0\) gives (1). For the length functional, we have

\[\begin{aligned} \dod{}{s} L(s) &= \int_a^b \dpd{}{s} \left\langle \dpd{\Gamma}{t} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle^{\frac12} \dif t \\ &= \int_a^b \left\langle \nabla_{\partial/\partial s} \dpd{\Gamma}{t} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle \left\langle \dpd{\Gamma}{t} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle^{-\frac12} \dif t \\ &= \int_a^b \left\langle \nabla_{\partial/\partial t} \dpd{\Gamma}{s} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle \left\langle \dpd{\Gamma}{t} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle^{-\frac12} \dif t \\ &= \int_a^b \left( \dpd{}{t} \left\langle \dpd{\Gamma}{s} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle - \left\langle \dpd{\Gamma}{s} (s,t), \nabla_{\partial/\partial t} \dpd{\Gamma}{t} (s,t) \right\rangle \right) \left\langle \dpd{\Gamma}{t} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle^{-\frac12} \dif t ~, \end{aligned}\]

The same interchange of derivatives and integration by parts applies to length, with the speed factor from differentiating the square root. Evaluating the last line at \(s=0\) gives (1), as desired. ◻

Remark 9. Relaxing the smooth curve assumption in Proposition 8 to \(c\) being piecewise differentiable, then the formulae include additional terms. Namely, let

\[\dot c(t_i^+)=\lim_{t\downarrow t_i}\partial_t\Gamma(0,t),\qquad \dot c(t_i^-)=\lim_{t\uparrow t_i}\partial_t\Gamma(0,t).\]

For a piecewise smooth curve, perform the integration by parts separately on each closed interval \([t_i,t_{i+1}]\), to obtain

\[\begin{aligned} \dod{}{s} E(s) &= \sum_{0 \leq i \leq k} \left\langle \dpd{\Gamma}{s} (s,t), \dpd{\Gamma}{t} (s,t) \right\rangle\bigg|_{t_i}^{t_{i+1}} - \int_a^b \left\langle \dpd{\Gamma}{s} (s,t), \nabla_{\partial/\partial t} \dpd{\Gamma}{t} (s,t) \right\rangle \dif t ~, \end{aligned}\]

which, evaluating at \(s=0\), gives

\[\begin{aligned} E'(0) &= - \int_a^b \left\langle V(t), \nabla_{\partial/\partial t} \dpd{\Gamma}{t} \right\rangle \dif t - \sum_{1 \leq i \leq k} \left\langle V(t_i), \dod{c}{t} (t_i^+) - \dod{c}{t} (t_i^{-}) \right\rangle \\ &\quad + \left\langle \dpd{\Gamma}{t} (0,b), V(b) \right\rangle - \left\langle \dpd{\Gamma}{t} (0,a), V(a) \right\rangle~. \end{aligned}\]

This is the first variation formula in (Carmo 1992).

The analogous piecewise-regular length formula is

\[L'(0)=\langle V,T\rangle\big|_a^b-\sum_i\langle V(t_i),T(t_i^+)-T(t_i^-)\rangle-\sum_i\int_{t_i}^{t_{i+1}}\langle V,D_tT\rangle\,dt,\]

where each one-sided speed is nonzero and \(T=\dot c/|\dot c|\) on that piece. The jump terms come from the upper endpoint of one integration interval and the lower endpoint of the next.

For energy with fixed endpoints, vanishing first variation for every continuous piecewise smooth \(V\) implies \(D_t\dot c=0\) on each open piece, by testing with fields supported inside that piece. The remaining sum then tests the velocity jumps. A continuous field can be prescribed arbitrarily at any single joining time and made zero at the other joining times, so each jump must vanish. ODE uniqueness joins these segments into one smooth geodesic. Conversely a geodesic has zero acceleration and no jumps, making the proper first variation zero. Thus the piecewise formula proves the full critical-curve characterization rather than assuming absence of corners.

Carmo, Manfredo Perdigão do. 1992. Riemannian Geometry. Mathematics. Theory & Applications. Boston: Birkhäuser.

Reading: Carmo (1992), Riemannian geometry.