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Anton Petrunin
Anton Petrunin
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Interesting question. The answer is no: surfaces with bounded geometry can have normal bundles with unbounded curvature.

To set the stage, it's worth first noting that you can have a surface with extreme geometry isometrically embedded in $\mathbb E^3$, where the normal bundle, being one-dimensional, has a trivial connection; or include this into $\mathbb E^4$ (Euclidean 4-space) where the normal bundle is 2-dimensional, but the curvature is still 0. This at least illustrates that bounded geometry of the normal bundle and tangent bundle are decoupled.

I'll now describe an isometric embedding of the $\mathbb E^2$ into $\mathbb E^6$ where the connection on the normal bundle has unbounded curvature. The embedding have local 1-parameter groups of symmetry, which makes it easier to keep track of curvature without needing to write down equations.

Start by visualizing a helical curve in $\mathbb E^3$. The tangent vector to a helix goes repeatedly around a circle in its spherical image. The connection on the tangent bundle is induced by this Gauss map from the connection on $S^2$, so the parallel translation of the normal bundle rotates the plane by an angle equal to the area enclosed inside this circle once every coil of the helix.

Now consider a similar curve in $\mathbb E^5$, thought of as $\mathbb E \times \mathbb E^2 \times \mathbb E^2$. In the $\E$$\mathbb E$-direction, the curve makes uniform progress, while going around circles of possibly different radii at possibly different rates in the $\mathbb E^2$ directions. If the term weren't otherwise engaged, one could call this a double helix. It is invariant by a 1-parameter group of isometries of $\mathbb E^5$ that translates in the $\mathbb E$ direction while spinning the two perpendicular planes at their own rates.

The normal bundle splits into two $\mathbb E^2$ subbundles, its intersection with the two 3-dimensional $\mathbb E \times \mathbb E^2$'s. The connection preserves this splitting, rotating the two $\mathbb E^2$'s indendently.

Now add an extra "parameter" dimension, making the ambient space $\mathbb E^6$. Modify the curve in $\E^5$$\mathbb E^5$ by increasing the radius of one helix while decreasing the radius of the other, balancing the changes so the curve remains invariant by the same 1-parameter group, and its arc length remains constant (as measured by the time parameter of the 1-parameter group). It's easy to see, since it's locally isometric to a surface of revolution because of the symmetry, that the resulting surface is isometric to $\mathbb E^2$. We can make the circle in on $\mathbb E^2$ go all the way to 0. After making sure it has $C^\infty$ contact to a straight line in this projection, we can then start making this projection helical again, but with a steeper and tighter helix, adjusting by letting the other helical projection shrink to a line. The local symmetry group in $\mathbb E^6$ has changed, but the induced symmetry on the surface remains the same. We can go back and forth, alternating between helical effects in the two factors, inexorably tightening the screws without distoring the surface.

The curvature of the induced connection on the normal bundle becomes arbitrarily high, as you can see by following the connection around a small rectangle, with two edges in the "parameter" direction, one edge where say the first helix has become straight and the fourth edge where the first helix is wound in very small tight coils.

Interesting question. The answer is no: surfaces with bounded geometry can have normal bundles with unbounded curvature.

To set the stage, it's worth first noting that you can have a surface with extreme geometry isometrically embedded in $\mathbb E^3$, where the normal bundle, being one-dimensional, has a trivial connection; or include this into $\mathbb E^4$ (Euclidean 4-space) where the normal bundle is 2-dimensional, but the curvature is still 0. This at least illustrates that bounded geometry of the normal bundle and tangent bundle are decoupled.

I'll now describe an isometric embedding of the $\mathbb E^2$ into $\mathbb E^6$ where the connection on the normal bundle has unbounded curvature. The embedding have local 1-parameter groups of symmetry, which makes it easier to keep track of curvature without needing to write down equations.

Start by visualizing a helical curve in $\mathbb E^3$. The tangent vector to a helix goes repeatedly around a circle in its spherical image. The connection on the tangent bundle is induced by this Gauss map from the connection on $S^2$, so the parallel translation of the normal bundle rotates the plane by an angle equal to the area enclosed inside this circle once every coil of the helix.

Now consider a similar curve in $\mathbb E^5$, thought of as $\mathbb E \times \mathbb E^2 \times \mathbb E^2$. In the $\E$-direction, the curve makes uniform progress, while going around circles of possibly different radii at possibly different rates in the $\mathbb E^2$ directions. If the term weren't otherwise engaged, one could call this a double helix. It is invariant by a 1-parameter group of isometries of $\mathbb E^5$ that translates in the $\mathbb E$ direction while spinning the two perpendicular planes at their own rates.

The normal bundle splits into two $\mathbb E^2$ subbundles, its intersection with the two 3-dimensional $\mathbb E \times \mathbb E^2$'s. The connection preserves this splitting, rotating the two $\mathbb E^2$'s indendently.

Now add an extra "parameter" dimension, making the ambient space $\mathbb E^6$. Modify the curve in $\E^5$ by increasing the radius of one helix while decreasing the radius of the other, balancing the changes so the curve remains invariant by the same 1-parameter group, and its arc length remains constant (as measured by the time parameter of the 1-parameter group). It's easy to see, since it's locally isometric to a surface of revolution because of the symmetry, that the resulting surface is isometric to $\mathbb E^2$. We can make the circle in on $\mathbb E^2$ go all the way to 0. After making sure it has $C^\infty$ contact to a straight line in this projection, we can then start making this projection helical again, but with a steeper and tighter helix, adjusting by letting the other helical projection shrink to a line. The local symmetry group in $\mathbb E^6$ has changed, but the induced symmetry on the surface remains the same. We can go back and forth, alternating between helical effects in the two factors, inexorably tightening the screws without distoring the surface.

The curvature of the induced connection on the normal bundle becomes arbitrarily high, as you can see by following the connection around a small rectangle, with two edges in the "parameter" direction, one edge where say the first helix has become straight and the fourth edge where the first helix is wound in very small tight coils.

Interesting question. The answer is no: surfaces with bounded geometry can have normal bundles with unbounded curvature.

To set the stage, it's worth first noting that you can have a surface with extreme geometry isometrically embedded in $\mathbb E^3$, where the normal bundle, being one-dimensional, has a trivial connection; or include this into $\mathbb E^4$ (Euclidean 4-space) where the normal bundle is 2-dimensional, but the curvature is still 0. This at least illustrates that bounded geometry of the normal bundle and tangent bundle are decoupled.

I'll now describe an isometric embedding of the $\mathbb E^2$ into $\mathbb E^6$ where the connection on the normal bundle has unbounded curvature. The embedding have local 1-parameter groups of symmetry, which makes it easier to keep track of curvature without needing to write down equations.

Start by visualizing a helical curve in $\mathbb E^3$. The tangent vector to a helix goes repeatedly around a circle in its spherical image. The connection on the tangent bundle is induced by this Gauss map from the connection on $S^2$, so the parallel translation of the normal bundle rotates the plane by an angle equal to the area enclosed inside this circle once every coil of the helix.

Now consider a similar curve in $\mathbb E^5$, thought of as $\mathbb E \times \mathbb E^2 \times \mathbb E^2$. In the $\mathbb E$-direction, the curve makes uniform progress, while going around circles of possibly different radii at possibly different rates in the $\mathbb E^2$ directions. If the term weren't otherwise engaged, one could call this a double helix. It is invariant by a 1-parameter group of isometries of $\mathbb E^5$ that translates in the $\mathbb E$ direction while spinning the two perpendicular planes at their own rates.

The normal bundle splits into two $\mathbb E^2$ subbundles, its intersection with the two 3-dimensional $\mathbb E \times \mathbb E^2$'s. The connection preserves this splitting, rotating the two $\mathbb E^2$'s indendently.

Now add an extra "parameter" dimension, making the ambient space $\mathbb E^6$. Modify the curve in $\mathbb E^5$ by increasing the radius of one helix while decreasing the radius of the other, balancing the changes so the curve remains invariant by the same 1-parameter group, and its arc length remains constant (as measured by the time parameter of the 1-parameter group). It's easy to see, since it's locally isometric to a surface of revolution because of the symmetry, that the resulting surface is isometric to $\mathbb E^2$. We can make the circle in on $\mathbb E^2$ go all the way to 0. After making sure it has $C^\infty$ contact to a straight line in this projection, we can then start making this projection helical again, but with a steeper and tighter helix, adjusting by letting the other helical projection shrink to a line. The local symmetry group in $\mathbb E^6$ has changed, but the induced symmetry on the surface remains the same. We can go back and forth, alternating between helical effects in the two factors, inexorably tightening the screws without distoring the surface.

The curvature of the induced connection on the normal bundle becomes arbitrarily high, as you can see by following the connection around a small rectangle, with two edges in the "parameter" direction, one edge where say the first helix has become straight and the fourth edge where the first helix is wound in very small tight coils.

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Bill Thurston
Bill Thurston
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Interesting question. The answer is no: surfaces with bounded geometry can have normal bundles with unbounded curvature.

To set the stage, it's worth first noting that you can have a surface with extreme geometry isometrically embedded in $\mathbb E^3$, where the normal bundle, being one-dimensional, has a trivial connection; or include this into $\mathbb E^4$ (Euclidean 4-space) where the normal bundle is 2-dimensional, but the curvature is still 0. This at least illustrates that bounded geometry of the normal bundle and tangent bundle are decoupled.

I'll now describe an isometric embedding of the $\mathbb E^2$ into $\mathbb E^6$ where the connection on the normal bundle has unbounded curvature. The embedding have local 1-parameter groups of symmetry, which makes it easier to keep track of curvature without needing to write down equations.

Start by visualizing a helical curve in $\mathbb E^3$. The tangent vector to a helix goes repeatedly around a circle in its spherical image. The connection on the tangent bundle is induced by this Gauss map from the connection on $S^2$, so the parallel translation of the normal bundle rotates the plane by an angle equal to the area enclosed inside this circle once every coil of the helix.

Now consider a similar curve in $\mathbb E^5$, thought of as $\mathbb E \times \mathbb E^2 \times \mathbb E^2$. In the $\E$-direction, the curve makes uniform progress, while going around circles of possibly different radii at possibly different rates in the $\mathbb E^2$ directions. If the term weren't otherwise engaged, one could call this a double helix. It is invariant by a 1-parameter group of isometries of $\mathbb E^5$ that translates in the $\mathbb E$ direction while spinning the two perpendicular planes at their own rates.

The normal bundle splits into two $\mathbb E^2$ subbundles, its intersection with the two 3-dimensional $\mathbb E \times \mathbb E^2$'s. The connection preserves this splitting, rotating the two $\mathbb E^2$'s indendently.

Now add an extra "parameter" dimension, making the ambient space $\mathbb E^6$. Modify the curve in $\E^5$ by increasing the radius of one helix while decreasing the radius of the other, balancing the changes so the curve remains invariant by the same 1-parameter group, and its arc length remains constant (as measured by the time parameter of the 1-parameter group). It's easy to see, since it's locally isometric to a surface of revolution because of the symmetry, that the resulting surface is isometric to $\mathbb E^2$. We can make the circle in on $\mathbb E^2$ go all the way to 0. After making sure it has $C^\infty$ contact to a straight line in this projection, we can then start making this projection helical again, but with a steeper and tighter helix, adjusting by letting the other helical projection shrink to a line. The local symmetry group in $\mathbb E^6$ has changed, but the induced symmetry on the surface remains the same. We can go back and forth, alternating between helical effects in the two factors, inexorably tightening the screws without distoring the surface.

The curvature of the induced connection on the normal bundle becomes arbitrarily high, as you can see by following the connection around a small rectangle, with two edges in the "parameter" direction, one edge where say the first helix has become straight and the fourth edge where the first helix is wound in very small tight coils.