MathDB

Let

H

be the hyperbolic plane,

I(H)

be the isometry group of

H

, and

O\in H

be a fixed starting point. Determine those continuous

\sigma\colon H\rightarrow I(H)

mappings that satisfty the following three conditions: (a)

\sigma(O)=\mathrm{id}

, and

\sigma (X)O=X

for all

X\in H

; (b) for every

X\in H\backslash \{ O\}

point, the

\sigma(X)

isometry is a paracyclic shift, i.e. every member of a system of paracycles through a common infinitely far point is left invariant; (c) for any pair

P,Q\in H

of points there exists a point

X\in H

such that

\sigma(X)P=Q

. Prove that the

\sigma\colon H\rightarrow I(H)

mappings satisfying the above conditions are differentiable with the exception of a point.

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