Null-homotopy versus shrinkability through short loops on $S^2$ and on a short circle

The two-sphere (i). $S^2$ is CAT(1) by [L1], so by [L3] every isometrically embedded circle in $S^2$ has length at least $2\pi$, i.e. $m\ge 2\pi$; the equator is an isometrically embedded circle of length $2\pi$, so $m=2\pi$, and L3 gives that every loop of length $<2\pi$ is shrinkable.

The latitude homotopy (ii). Parametrize the parallel by $E_{\phi }(t)=(\cos \phi \cos (2\pi t),\cos \phi \sin (2\pi t),\sin \phi )$. For an angular increment $u$ with $\mathrm{\mid }u\mathrm{\mid }\le \pi$, dot products and the addition formulas give the spherical distance $D_{\phi }(u)=2\arcsin (\cos \phi \sin (\mathrm{\mid }u\mathrm{\mid }\mathrm{/}2))$. Its right derivative at zero is $\cos \phi$, by The derivatives of sine and cosine are cosine and minus sine and For $-1<y<1$, $(\arcsin y{)}^{\mathrm{\prime }}=1\mathrm{/}\sqrt{1-y^2}$ and $(\arccos y{)}^{\mathrm{\prime }}=-1\mathrm{/}\sqrt{1-y^2}$. Consequently, for every $ϵ>0$, all sufficiently small increments satisfy $(\cos \phi -ϵ)\mathrm{\mid }u\mathrm{\mid }\le D_{\phi }(u)\le (\cos \phi +ϵ)\mathrm{\mid }u\mathrm{\mid }$. Refining any partition and summing proves that its supremum length on an angular interval of size $A$ is $A\cos \phi$, using the metric partition definition Length in a metric target: lower semicontinuity and arc-length reparametrization. Thus $L(E_{\phi })=2\pi \cos \phi$ and these loops are normalized, including the constant pole. The displayed coordinates vary uniformly continuously with $\phi$, and their lengths vary continuously. This gives the asserted null-homotopy, with short members for $\phi >0$, while $E_0$ has length $2\pi$ and is outside the domain of short-loop homotopy.

The short circle (iii). In $S_{\mathrm{\ell }}^1$, every pair at distance $<\mathrm{\ell }\mathrm{/}2$ has exactly one shortest arc, whereas opposite points have two distinct minimizing arcs of length $\mathrm{\ell }\mathrm{/}2$. Any isometrically embedded circle of length $u$ has two minimizing arcs between its opposite points, so $u\mathrm{/}2\ge \mathrm{\ell }\mathrm{/}2$. The full circle realizes equality, proving $m=\mathrm{\ell }$. By [L3] every loop of length $<\mathrm{\ell }$ is shrinkable; the full circle is nonshrinkable. For the ordinary null-homotopy assertions, lift a loop to $\mathbb{R}$ under $t\mapsto t<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\mathrm{\ell }$: subdivision into arcs lying in intervals of length $<\mathrm{\ell }\mathrm{/}2$ gives successive unique local lifts once the initial value is fixed. The endpoint displacement is $k\mathrm{\ell }$ for an integer $k$. A loop of length $<\mathrm{\ell }$ has $\mathrm{\mid }k\mathrm{\mid }\mathrm{\ell }\le L<\mathrm{\ell }$, so $k=0$ and its lift is closed; For any loop with $k=0$, multiplying its closed lift about its initial point by $t\in [0,1]$ contracts it through loops of lengths $tL$: local lifts preserve length by the partition definition, and Euclidean scaling multiplies length by $t$. For a normalized loop this family is normalized and continuous in the uniform-plus-length topology, including $t=0$. Thus every short zero-winding loop is shrinkable. For a continuous homotopy, compact uniform continuity [L4] gives a common finite subdivision into the same local lifting charts near each parameter value; compatible local lifts therefore depend continuously on the parameter. Their endpoint displacement is a continuous integer multiple of $\mathrm{\ell }$, hence constant on the parameter interval. The full circle has $k=1$ and a constant loop has $k=0$, so the full circle is not null-homotopic. Thus every nonzero-winding short loop is nonshrinkable, has length at least $\mathrm{\ell }$, and is not null-homotopic, proving the asserted classification.

Separation (iv). In a compact geodesic locally CAT(1) space the existence of a short nonshrinkable loop is equivalent to the failure of CAT(1) by L3, and when $m<2\pi$ the minimum nonshrinkable length is $m$, realized by an isometrically embedded circle; both examples above are instances of this criterion, since $S^2$ has $m=2\pi$ and no short nonshrinkable loop, while $S_{\mathrm{\ell }}^1$ has $m=\mathrm{\ell }<2\pi$ and the full circle as shortest nonshrinkable loop.

Conclusion. Clause (i) is step 1.1, clause (ii) is step 1.2, clause (iii) is step 1.3 and clause (iv) is step 1.4; AC enters only through the supplier [L3] ([L6]).