From Amsler Surface to Pseudosphere: Journey via a 6‑Dimensional Extended TSO

We present a unified geometric framework that continuously deforms the symmetric Amsler surface into the pseudosphere (kink solution). The construction uses a six‑dimensional extension of the Time‑Shared Object (TSO) that incorporates a branch parameter selecting the solution of the reduced ODE and an orientation angle for the degenerate line. The GAE family with the \(\pi s\) condition provides a path from the Amsler surface to the constant \(2\pi\) solution; at this endpoint the surface collapses to a horizontal line lying in the \((u,v)\)-plane. A one‑parameter family of pendulum solutions connects the same constant \(2\pi\) solution to the kink, but its degenerate limit is the vertical \(z\)-axis. To join the two families at the level of immersed surfaces, we insert an explicit rigid rotation that continuously rotates the horizontal line into the vertical line. By smoothly concatenating the four segments we obtain a \(C^{\infty}\) path in the extended space \(\mathcal{E} = \mathcal{T}_5 \times S^1\), demonstrating that the two classical pseudospherical surfaces are continuously deformable into one another. The paper details the limiting behaviour at \(s=0,1,2,3\), the construction of the 5‑dimensional TSO \(\mathcal{T}_5\), the degenerate metric connections, the rotation step, and the final smooth path.