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The answer to the first is NO.

As mentioned by another answer you can use Do Carmo to prove a local result of the type: at points where the normal curvature is non-zero the 0-directions defines a foliation of straight lines.

With this one can prove that embedding $\mathbb{R}²$ into $\mathbb{R}^3$ preserves the diameter of circles, but one can construct a counter example in the case of a piece of paper:

Take a equilateral triangle on your table with points barely out side of the piece of paper. Then since these sides do not meet inside the piece you may fold (using very sharp foldings) the piece of paper up at the sides of the triangle such that out side of a nighborhood of the triangle the paper lies in planes which are perpendicular to the table. Note that this can not be extended to all of $\mathbb{R}^2$ because the folding lines interset! Then shrinking the circumscribed sphere slightly of the triangle to lie in the paper - this is folded up so that the diameter becomes strictly less.

For the second question I really dont see $\pi_1$ any where. What I see is the following: for me it is natural to defined the distance on a Riemmanian manifold $X$ as:

$d(x,y) = \inf_{\gamma} \textrm{len}(\gamma)$

where the infimum is over curves $\gamma$ from $x$ to $y$, and len($\gamma$) is the length of the curve. This is intrinsic and gives the same diameter diameters for the piece of paper, but when you removed remove an open set it becomes different, because the curves has to avoid this set.

Since isometric embeddings preserve lengths of curves you get for my with this definition

$d(Fx,Fy) \leq d(x,y)$

3 added 217 characters in body; deleted 1 characters in body

The answer to the first is NO.

As mentioned by another answer you can use Do Carmo to prove a local result of the type: at points where the normal curvature is non-zero the 0-directions defines a foliation of straight lines.

With this one can prove that embedding $\mathbb{R}²$ into $\mathbb{R}^3$ preserves the diameter of circles, but one can construct a counter example in the case of a piece of paper:

Take a equilateral triangle on your table with points barely out side of the piece of paper. Then since these sides do not meet inside the piece you may fold (using very sharp foldings) the piece of paper up at the sides of the triangle such that out side of a nighborhood of the triangle the paper lies in planes which are perpendicular to the table. Note that this can not be extended to all of $\mathbb{R}^2$ because the folding lines interset! Then shrinking the circumscribed sphere slightly of the triangle to lie in the paper - this is folded up so that the diameter becomes strictly less.

For the second question I really dont see $\pi_1$ any wherejust the fact that removing an open subset in the interior increases the diameter of . What I see is the circles before they are embedded into $\mathbb{R}^3$. This following: for me it is so because natural to defined the distance on a Riemannian Riemmanian manifold is intrinsically given by $X$ as:

$d(x,y) = \inf_{\gamma} \textrm{len}(\gamma)$

where the infimum is over curves $\gamma$ from $x$ to $y$, and len($\gamma$) is the length of curves connecting the pointscurve. This is intrinsic and gives the same diameter for the piece of paper, but when you removed an open set it becomes different.

Since isometric embeddings preserve lengths of curves you get for my definition

$d(Fx,Fy) \leq d(x,y)$

2 deleted 123 characters in body

The answer to the first is NO.

As mentioned by another answer you can use Do Carmo to prove a local result of the type: Locally the image of at points where the embedding normal curvature is up to an isometry of $\mathbb{R}^3$ given by non-zero the 0-directions defines a product foliation of an isometric embedding $(\epsilon,\epsilon) \to \mathbb{R}^2$ and the usual inclusion $(\epsilon,\epsilon)\subset \mathbb{R}$.straight lines.

With this in mind it is clear one can prove that embedding $\mathbb{R}²$ into $\mathbb{R}^3$ preserves the diameter of circles, but one can construct a counter example in the case of a piece of paper:

Take a equilateral triangle on your table with points barely out side of the piece of paper. Then since these sides do not meet inside the piece you may fold (using very sharp foldings) the piece of paper up at the sides of the triangle such that out side of a nighborhood of the triangle the paper lies in planes which are perpendicular to the table. Note that this can not be extended to all of $\mathbb{R}^2$ because the folding lines interset! Then shrinking the circumscribed sphere slightly of the triangle to lie in the paper - this is folded up so that the diameter becomes strictly less.

For the second question I really dont see $\pi_1$ any where just the fact that removing an open subset in the interior increases the diameter of the circles before they are embedded into $\mathbb{R}^3$. This is so because the distance on a Riemannian manifold is intrinsically given by infimum over length of curves connecting the points.

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