- #1

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## Main Question or Discussion Point

Let [itex]G\subset\mathbb{R}^n[/itex] be some open set, and [itex]x_1, x_2:[0,1]\to G[/itex] be differentiable paths with the same starting and ending points. Assume that there exists a homotopy [itex]f:[0,1]^2\to G[/itex] between the two paths. That means that the f is continuous, and the following conditions hold.

[tex]

f(t,0)=x_1(t),\quad f(t,1)=x_2(t),\quad f(0,s) = x_1(0) = x_2(0),\quad f(1,s) = x_1(1) = x_2(1)

[/tex]

How do you prove, that there also exists a homotopy [itex]g:[0,1]^2\to G[/itex] between the two paths so, that for all [itex]s\in [0,1][/itex], the mapping

[tex]

t\mapsto g(t,s)

[/tex]

is differentiable?

This seems a clear claim, but doesn't seem to come easily from the definitions.

[tex]

f(t,0)=x_1(t),\quad f(t,1)=x_2(t),\quad f(0,s) = x_1(0) = x_2(0),\quad f(1,s) = x_1(1) = x_2(1)

[/tex]

How do you prove, that there also exists a homotopy [itex]g:[0,1]^2\to G[/itex] between the two paths so, that for all [itex]s\in [0,1][/itex], the mapping

[tex]

t\mapsto g(t,s)

[/tex]

is differentiable?

This seems a clear claim, but doesn't seem to come easily from the definitions.