Weight-one regular axis gives elementary flatness of small axial orbits

Fix a chart $x:U\to {\mathbb{R}}^4$ with $x(p)=0$ and an open coordinate box $B$ about $0$ whose closure lies in $x(U)$. For each $s\in [0,2\pi ]$, joint continuity and ${\psi }_s(p)=p$ give a product neighborhood of $(s,0)$ mapped into $x^{-1}(B)$. A finite subcover of $[0,2\pi ]$ gives $ϵ>0$ such that $F(s,\rho ):=x({\psi }_s(\gamma (\rho )))$ is smooth on a neighborhood of $K=[0,2\pi ]\times [0,ϵ]$ and takes values in $B$ there, after shrinking $ϵ$. Put $X={\mathrm{\partial }}_{s}F$. Then $F(s,0)=0$, $X(s,0)=0$, and the chain rule gives ${\mathrm{\partial }}_{\rho }F(s,0)=d{x}_p(R_{s}v)$. If $J$ is the unit quarter-turn generator on $N_p$, commuting the mixed derivatives gives ${\mathrm{\partial }}_{\rho }X(s,0)=d{x}_p(R_{s}Jv)$.

The period interval counts one orbit. Indeed, pulling back the connection by an isometry preserves torsion freedom and metric compatibility, so uniqueness makes that pullback the original connection. Isometries therefore take geodesics to geodesics, and uniqueness of their initial-value solutions gives ${\psi }_s({\exp }_{p}w)={\exp }_p(D_p{\psi }_{s}w)$ whenever these expressions are defined. Choose an auxiliary positive definite norm on $T_{p}A$, combine it with $g_p{\mathrm{\mid }}_{N_p}$ on the direct sum, and take a sufficiently small ball $W$ for this norm inside the exponential diffeomorphism neighborhood. The identity-plus-rotation action preserves $W$, so both exponential expressions lie in its injectivity domain for every $s$. For small $\rho$, put $w(\rho )=({\exp }_p{\mathrm{\mid }}_W{)}^{-1}(\gamma (\rho ))$. The inverse differential at $p$ is the identity by the chain rule; hence $w(0)=0$ and $w^{\mathrm{\prime }}(0)=v$. Taylor's formula and a bound on $w^{\mathrm{\prime }\mathrm{\prime }}$ on a small compact interval give $w(\rho )=\rho v+O({\rho }^2)$. Its normal component is nonzero for small positive $\rho$. Since the tangent component is fixed and a weight-one rotation fixes a nonzero normal vector only for $s\in 2\pi \mathbb{Z}$, injectivity of ${\exp }_p{\mathrm{\mid }}_W$ shows that ${\psi }_s(\gamma (\rho ))=\gamma (\rho )$ only for such $s$.

The continuous components of ${\mathrm{\partial }}_{\rho }^{2}X$ are bounded on $K$ by compactness. Taylor's formula on each fixed-$s$ segment therefore gives $X(s,\rho )=\rho d{x}_p(R_{s}Jv)+O({\rho }^2)$ with a constant independent of $s$. To apply the stated open-convex-set version, each scalar component is restricted to an open interval about $[0,ϵ]$ at that fixed $s$; its remainder evaluation point lies in $K$, where the same bound applies. The same bounded-derivative argument gives $F(s,\rho )=O(\rho )$ uniformly. The first and second coordinate derivatives of the metric components are bounded on $\bar{B}$, so Taylor's formula along the coordinate segment from $0$ to $F(s,\rho )$ gives $g_{ab}(F(s,\rho ))=g_{ab}(0)+O(\rho )$ uniformly. The leading vectors $R_{s}Jv$ are uniformly bounded in these coordinates and have $g_p(R_{s}Jv,R_{s}Jv)=1$. Consequently the squared orbit speed $q(s,\rho ):=g_{{\psi }_s(\gamma (\rho ))}({\mathrm{\partial }}_s{\psi }_s(\gamma (\rho )),{\mathrm{\partial }}_s{\psi }_s(\gamma (\rho )))$ satisfies $q(s,\rho )={\rho }^2+O({\rho }^3)$ uniformly for $s\in [0,2\pi ]$ and is positive for sufficiently small $\rho >0$.

Write the squared orbit speed as $q(s,\rho )={\rho }^2(1+e(s,\rho ))$, with $\mathrm{\mid }e(s,\rho )\mathrm{\mid }\le C\rho$ uniformly by step 2.1. For small positive $\rho$, $1+e>0$ and $\mathrm{\mid }\sqrt{1+e}-1\mathrm{\mid }=\mathrm{\mid }e\mathrm{\mid }\mathrm{/}(\sqrt{1+e}+1)\le \mathrm{\mid }e\mathrm{\mid }$. Thus its speed lies between $\rho -C{\rho }^2$ and $\rho +C{\rho }^2$. The length definition and the continuous-integral bounds give orbit length $2\pi \rho +O({\rho }^2)$, counted once by step 1.2. Along the transverse curve, the smooth scalar $h(t)=g_{\gamma (t)}({\gamma }^{\mathrm{\prime }}(t),{\gamma }^{\mathrm{\prime }}(t))$ has $h(0)=1$. Bounded derivatives on a small compact interval and Taylor's formula give $h(t)=1+O(t)$, which is positive there. The same square-root identity gives speed $1+O(t)$; on $0\le t\le \rho$ it therefore lies between $1-C^{\mathrm{\prime }}\rho$ and $1+C^{\mathrm{\prime }}\rho$. Integrating yields radial length $\rho +O({\rho }^2)$. Dividing the two positive lengths gives $(2\pi +O(\rho ))\mathrm{/}(1+O(\rho ))\to 2\pi$. This proves the stated local absence of a conical defect under the assumed smooth weight-one axis data, without deriving those data from global hyperbolicity.