First variation formula for length

On each closed central piece, $\mathrm{\mid }T{\mathrm{\mid }}_g$ is continuous and positive, so [F3] gives a positive minimum. Continuity of $g({\mathrm{\partial }}_t\alpha ,{\mathrm{\partial }}_t\alpha )$ on a small compact parameter rectangle and uniform continuity in [F3] then give a common $\delta >0$ such that ${\mathrm{\partial }}_t\alpha (s,t)\mathrm{\ne }0$ on every strip whenever $\mathrm{\mid }s\mathrm{\mid }\le \delta$. Thus the speed is smooth there and differentiation under its integral is legitimate.

Metric compatibility and the derivative of the positive square root give ${{\mathrm{\partial }}_s\mid }_0\mathrm{\mid }{\mathrm{\partial }}_t\alpha {\mathrm{\mid }}_g=\frac{g(D_s{\mathrm{\partial }}_t\alpha ,T)}{\mathrm{\mid }T{\mathrm{\mid }}_g}=g(D_s{\mathrm{\partial }}_t\alpha ,U)\mathrm{.}$ In local coordinates, equality of mixed partials and the symmetric lower Christoffel indices in [F2] give $D_s{\mathrm{\partial }}_t\alpha =D_t{\mathrm{\partial }}_s\alpha$. Hence [F3] yields ${\frac{d}{ds}\mid }_{0}L({\gamma }_s)={\sum }_{j=1}^m{\int }_{t_{j-1}}^{t_j}g(D_{t}V,U)dt\mathrm{.}$

On each smooth piece, metric compatibility says $g(D_{t}V,U)=\frac{d}{dt}g(V,U)-g(V,D_{t}U)\mathrm{.}$ Newton--Leibniz from [F3] therefore turns step 2.1 into the sum of $g(V(t_j),U(t_j^{-}))-g(V(t_{j-1}),U(t_{j-1}^{+}))$ minus the displayed integrals of $g(V,D_{t}U)$.

Continuity of $V$ telescopes the interior boundary values to $-g(V(t_j),U(t_j^{+})-U(t_j^{-}))$, while the two surviving outer terms have the signs stated. If $m=1$ the corner sum is empty; if endpoints are fixed, [F1] gives $V(a)=V(b)=0$. When $\mathrm{\mid }T{\mathrm{\mid }}_g=1$ on every piece, $U=T$, so this formula agrees term by term with [F4].

Regularity excludes a constant or zero-length central curve on $a<b$ and excludes all such curves in dimension zero; those are genuinely outside the theorem rather than hidden divisions by zero. A zero variation field makes the derivative and all terms zero. Dimension one is unchanged. An empty target admits no given curve. One-sided endpoint and corner velocities are explicit in the statement, and compactness is used only over finitely many supplied strips. The Levi--Civita connection is unique and every compactness argument in [F3] is choice-free, so no choice axiom is used.