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rpotrie
rpotrie
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One possible way is to demand that the target set $f(p)$ is a compact set and use Hausdorff distance on the set of compact subsets of $N$ (this aplies to your application by considering the proyective space of $T_p M$).

In the particular aplication, I would useIf the fact that an open convex cone of $T_p M$ is defined by a subspace of $T_p M$ together with an angle. So, if and we assume that continuity implies that the dimension of the cones is constant (this would be given using the Hausdorff distance in the proyective space as above), you can test "smoothness" by considering the map from $M$ to the bundle of grasmannians times angle which is a diferentiable manifold and gives a meaningful way of saying that $f$ is smooth.

However, I believe it should be accepted that the cones "colapse" and decrease its dimension (even if they are never trivial, they can "change their dimension"). For this, the only way I imagine is to consider subsets $(T_pM)^n \times \mathbb{R}$ and consider the cone as the one generated by the $n$ vectors and with angle the value in $\mathbb{R}$ (clearly, there is no canonical way of considering this function, one should say that the map is smooth if there exists a function $g$ to $(T_pM)^n \times \mathbb{R}$ which defines $f$).

One possible way is to demand that the target set $f(p)$ is a compact set and use Hausdorff distance on the set of compact subsets of $N$ (this aplies to your application by considering the proyective space of $T_p M$).

In the particular aplication, I would use the fact that an open convex cone of $T_p M$ is defined by a subspace of $T_p M$ together with an angle. So, if we assume that continuity implies that the dimension of the cones is constant (this would be given using the Hausdorff distance in the proyective space as above), you can test "smoothness" by considering the map from $M$ to the bundle of grasmannians times angle which is a diferentiable manifold and gives a meaningful way of saying that $f$ is smooth.

However, I believe it should be accepted that the cones "colapse" and decrease its dimension. For this, the only way I imagine is to consider subsets $(T_pM)^n \times \mathbb{R}$ and consider the cone as the one generated by the $n$ vectors and with angle the value in $\mathbb{R}$ (clearly, there is no canonical way of considering this function, one should say that the map is smooth if there exists a function $g$ to $(T_pM)^n \times \mathbb{R}$ which defines $f$).

One possible way is to demand that the target set $f(p)$ is a compact set and use Hausdorff distance on the set of compact subsets of $N$ (this aplies to your application by considering the proyective space of $T_p M$).

In the particular aplication, If the cone of $T_p M$ is defined by a subspace of $T_p M$ together with an angle and we assume that continuity implies that the dimension of the cones is constant (this would be given using the Hausdorff distance in the proyective space as above), you can test "smoothness" by considering the map from $M$ to the bundle of grasmannians times angle which is a diferentiable manifold and gives a meaningful way of saying that $f$ is smooth.

However, I believe it should be accepted that the cones "colapse" and decrease its dimension (even if they are never trivial, they can "change their dimension"). For this, the only way I imagine is to consider subsets $(T_pM)^n \times \mathbb{R}$ and consider the cone as the one generated by the $n$ vectors and with angle the value in $\mathbb{R}$ (clearly, there is no canonical way of considering this function, one should say that the map is smooth if there exists a function $g$ to $(T_pM)^n \times \mathbb{R}$ which defines $f$).

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rpotrie
rpotrie
  • 3.9k
  • 24
  • 27

One possible way is to demand that the target set $f(p)$ is a compact set and use Hausdorff distance on the set of compact subsets of $N$ (this aplies to your application by considering the proyective space of $T_p M$).

In the particular aplication, I would use the fact that an open convex cone of $T_p M$ is defined by a subspace of $T_p M$ together with an angle. So, if we assume that continuity implies that the dimension of the cones is constant (this would be given using the Hausdorff distance in the proyective space as above), you can test "smoothness" by considering the map from $M$ to the bundle of grasmannians times angle which is a diferentiable manifold and gives a meaningful way of saying that $f$ is smooth.

However, I believe it should be accepted that the cones "colapse" and decrease its dimension. For this, the only way I imagine is to consider subsets $(T_pM)^n \times \mathbb{R}$ and consider the cone as the one generated by the $n$ vectors and with angle the value in $\mathbb{R}$ (clearly, there is no canonical way of considering this function, one should say that the map is smooth if there exists a function $g$ to $(T_pM)^n \times \mathbb{R}$ which defines $f$).