H-one Riemannian curves and their half-energy

The one-dimensional weak-coordinate equivalence. [F3, F4, F5, given] Let $u$ be a real weak-$W^{1,2}$ coordinate class on an interval $(c,d)$, with weak derivative $h\in L^2(c,d)$. Because the interval has finite length, $h\in L^1$. Set $H(t)={\int }_c^{t}h(s)ds$. For a compactly supported smooth test $\phi$, Fubini [F3] on the integrable triangle gives ${\int }_c^{d}H(t){\phi }^{\mathrm{\prime }}(t)dt={\int }_c^{d}h(s)({\int }_s^d{\phi }^{\mathrm{\prime }}(t)dt)ds=-{\int }_c^{d}h(s)\phi (s)ds\mathrm{.}$ Thus $u-H$ has zero distributional derivative; [F4] makes it a constant distribution and identifies $u$ almost everywhere with the unique continuous primitive $C+H$. Conversely the same Fubini identity gives weak derivative $h$ for every such primitive, which is bounded by Cauchy--Schwarz [F5] and hence in $L^2(c,d)$. This proves the claimed equivalence without invoking a sharp absolute-continuity FTC that assumes Dependent Choice.

Continuous representatives and the interval bound. [F5, given] For a primitive $x(t)=x(c)+{\int }_c^{t}h$, [F5] gives $\mathrm{\mid }x(t)-x(s)\mathrm{\mid }\le \mathrm{\parallel }h{\mathrm{\parallel }}_{L^2(c,d)}\sqrt{\mathrm{\mid }t-s\mathrm{\mid }}\mathrm{.}$ It is continuous, has fixed endpoint traces, is absolutely continuous by the integral criterion, and is bounded on the compact interval. The same bound coordinatewise holds for a finite-dimensional vector primitive. In particular the coordinate $H^1$ topology embeds continuously in the uniform topology.

Smooth superposition and chart invariance. [F2, F5, step 1.2, given] Let $x$ be such a vector primitive and let $F(t,z)$ be smooth on a compact coordinate tube containing its graph. Extend $h=x^{\mathrm{\prime }}$ by zero to $\mathbb{R}$ and use [F2] to choose smooth $h_k\to h$ in $L^2$; put $x_k(t)=x(c)+{\int }_c^t{h}_k$. Step 1.2 makes $x_k\to x$ uniformly. For each smooth $x_k$, the ordinary chain rule gives $F(t,x_k(t))-F(c,x_k(c))={\int }_c^t({\mathrm{\partial }}_{s}F(s,x_k(s))+D_{z}F(s,x_k(s))h_k(s))ds\mathrm{.}$ Uniform boundedness and uniform continuity of the first derivatives of $F$ on a slightly larger compact tube, together with $L^2$ convergence of $h_k$, let both sides converge to the same identity for $x$. Its integrand belongs to $L^2$ because the derivatives of $F$ are bounded there. Applying this to smooth coordinate transitions proves that the defining property and the almost-everywhere chain rule are independent of charts; finite subdivisions may be refined without changing them.

Speed, energy, topology and boundaries. [F1, F2, F5, step 1.2, step 2.1, given] The coordinate velocity is in $L^2$ on each of the finitely many pieces by step 2.1. Smoothness and positive definiteness of the metric [F1] make $\mathrm{\mid }\dot{\alpha }{\mathrm{\mid }}_g$ a measurable $L^2$ function; it is also $L^1$ by [F5] on the finite interval. Under a chart change the velocity and metric transform together, so its norm and the two integrals in the definition are invariant. For a piecewise smooth curve the a.e. velocity is its ordinary velocity, so the integrals agree with [F1] and the published half-energy convention. Near a fixed smooth reference curve, finitely many smooth coordinate tubes and step 2.1 make their coordinate $H^1$ norms locally equivalent; step 1.2 then implies uniform closeness in the manifold metric. Dimension zero gives constant curves and zero energy; empty manifolds have no curve instance. All selections in this verification are finite except the countable approximation sequence supplied by [F2], which is covered by the stated Countable Choice.