Geometric significance of the differential of the Gauss map, $dN_p: T_p(S) rightarrow T_p(S)$, being a self...
i'm studying differential geometry for the first time and I just started reading about the gauss map and I've gotta say this stuff is pretty cool. The claim made in the title of this post is from page 142 of Do Carmo's Differential geometry book, where it is stated as "proposition 1". I have never taken a proper geometry class before and the text is starting to get to the point where the density of the calculus/linear algebra is making the geometric significance of the statements rather opaque. That being said, I feel that there is an 'aha' moment just around the corner with this one, and people here are usually pretty good at breaking things down for me so I thought i'd ask y'll this rather soft question.
Anyway, just to refresh your memory, I'm looking to understand the geometric significance of the differential of the Gauss map, $dN_p: T_p(S) rightarrow T_p(S)$, being a self adjoint linear map. (in particular, what is the geometric significance of this linear map being 'self adjoint'.)
Where $N: S rightarrow S^2$ is a map from the set of normal unit vectors of a regualer surface $S$ to the unit sphere $S^2$ and $T_p(S)$ is that tangent plane of an arbitrary $p in S$.
Also, note that $T_p(S)$ and $T_{N(p)}(S^2)$ are the same as vector spaces, which is why we can view $dN_p$ as a linear map from $T_p(S) rightarrow T_p(S)$
differential-geometry
asked Dec 27 '18 at 0:53
Michael Vaughan
790211
add a comment |
i'm studying differential geometry for the first time and I just started reading about the gauss map and I've gotta say this stuff is pretty cool. The claim made in the title of this post is from page 142 of Do Carmo's Differential geometry book, where it is stated as "proposition 1". I have never taken a proper geometry class before and the text is starting to get to the point where the density of the calculus/linear algebra is making the geometric significance of the statements rather opaque. That being said, I feel that there is an 'aha' moment just around the corner with this one, and people here are usually pretty good at breaking things down for me so I thought i'd ask y'll this rather soft question.
Anyway, just to refresh your memory, I'm looking to understand the geometric significance of the differential of the Gauss map, $dN_p: T_p(S) rightarrow T_p(S)$, being a self adjoint linear map. (in particular, what is the geometric significance of this linear map being 'self adjoint'.)
Where $N: S rightarrow S^2$ is a map from the set of normal unit vectors of a regualer surface $S$ to the unit sphere $S^2$ and $T_p(S)$ is that tangent plane of an arbitrary $p in S$.
Also, note that $T_p(S)$ and $T_{N(p)}(S^2)$ are the same as vector spaces, which is why we can view $dN_p$ as a linear map from $T_p(S) rightarrow T_p(S)$
differential-geometry
asked Dec 27 '18 at 0:53
Michael Vaughan
790211
1
The number $dN_p(u) cdot v$ is roughly the "rate of change of the unit normal vector at $p$, when $p$ is pushed in the direction of $u$, as perceived by the subspace $langle v rangle subset T_p S$ (the dot with $v$ is to be interpreted as projection)". By self-adjointness, this number is symmetric in $u$ and $v$, so rate of change of $N_p$ when $p$ is pushed a little in the direction of $u$, as felt in the direction of $v$ is equal to rate of change of $N_p$ when $p$ is pushed in the direction of $v$, as felt in the direction of $u$. At least this is how I understand the result.
– Balarka Sen
Dec 27 '18 at 4:55
1
More algebraically, this means $Bbb{II}_p(u, v) = dN_p(u) cdot v$ is a (positive-definite) symmetric billinear form on $S$, also known as the second fundamental form. This is an inner product on the tangent spaces of $S$, but not the standard one (coming from just taking $u cdot v$ in $Bbb R^3$). This is a more extrinsic object, and carries information about the embedding of the surface in $Bbb R^3$ rather than the surface itself.
– Balarka Sen
Dec 27 '18 at 5:03
Thank you for these posts, they were both very enlightening. In your first post you say that the dot product can be interpreted as "the rate of change of the unit normal vector at p, when p is pushed in the direction of u, as perceived by the subspace $<v>$. Thank you a lot for that. Some part of me wants to always consider u and v to be orthogonal vectors in the surface they parameterize, but this is definitely not true, obviously my geometric intuition needs developing.
– Michael Vaughan
Dec 27 '18 at 14:13
1
I'd add that, using the slightly more abstract language of Riemannian geometry, this essentially boils down to the statement that the Levi-Civita connection is torsion-free.
– Or Eisenberg
Dec 27 '18 at 18:50
add a comment |
i'm studying differential geometry for the first time and I just started reading about the gauss map and I've gotta say this stuff is pretty cool. The claim made in the title of this post is from page 142 of Do Carmo's Differential geometry book, where it is stated as "proposition 1". I have never taken a proper geometry class before and the text is starting to get to the point where the density of the calculus/linear algebra is making the geometric significance of the statements rather opaque. That being said, I feel that there is an 'aha' moment just around the corner with this one, and people here are usually pretty good at breaking things down for me so I thought i'd ask y'll this rather soft question.
Anyway, just to refresh your memory, I'm looking to understand the geometric significance of the differential of the Gauss map, $dN_p: T_p(S) rightarrow T_p(S)$, being a self adjoint linear map. (in particular, what is the geometric significance of this linear map being 'self adjoint'.)
Where $N: S rightarrow S^2$ is a map from the set of normal unit vectors of a regualer surface $S$ to the unit sphere $S^2$ and $T_p(S)$ is that tangent plane of an arbitrary $p in S$.
Also, note that $T_p(S)$ and $T_{N(p)}(S^2)$ are the same as vector spaces, which is why we can view $dN_p$ as a linear map from $T_p(S) rightarrow T_p(S)$
differential-geometry
asked Dec 27 '18 at 0:53
Michael Vaughan
790211
i'm studying differential geometry for the first time and I just started reading about the gauss map and I've gotta say this stuff is pretty cool. The claim made in the title of this post is from page 142 of Do Carmo's Differential geometry book, where it is stated as "proposition 1". I have never taken a proper geometry class before and the text is starting to get to the point where the density of the calculus/linear algebra is making the geometric significance of the statements rather opaque. That being said, I feel that there is an 'aha' moment just around the corner with this one, and people here are usually pretty good at breaking things down for me so I thought i'd ask y'll this rather soft question.
Anyway, just to refresh your memory, I'm looking to understand the geometric significance of the differential of the Gauss map, $dN_p: T_p(S) rightarrow T_p(S)$, being a self adjoint linear map. (in particular, what is the geometric significance of this linear map being 'self adjoint'.)
Where $N: S rightarrow S^2$ is a map from the set of normal unit vectors of a regualer surface $S$ to the unit sphere $S^2$ and $T_p(S)$ is that tangent plane of an arbitrary $p in S$.
Also, note that $T_p(S)$ and $T_{N(p)}(S^2)$ are the same as vector spaces, which is why we can view $dN_p$ as a linear map from $T_p(S) rightarrow T_p(S)$
differential-geometry
differential-geometry
asked Dec 27 '18 at 0:53
Michael Vaughan
790211
asked Dec 27 '18 at 0:53
Michael Vaughan
790211
asked Dec 27 '18 at 0:53
Michael Vaughan
790211
asked Dec 27 '18 at 0:53
Michael Vaughan
790211
asked Dec 27 '18 at 0:53
Michael Vaughan
790211
790211
1
The number $dN_p(u) cdot v$ is roughly the "rate of change of the unit normal vector at $p$, when $p$ is pushed in the direction of $u$, as perceived by the subspace $langle v rangle subset T_p S$ (the dot with $v$ is to be interpreted as projection)". By self-adjointness, this number is symmetric in $u$ and $v$, so rate of change of $N_p$ when $p$ is pushed a little in the direction of $u$, as felt in the direction of $v$ is equal to rate of change of $N_p$ when $p$ is pushed in the direction of $v$, as felt in the direction of $u$. At least this is how I understand the result.
– Balarka Sen
Dec 27 '18 at 4:55
1
More algebraically, this means $Bbb{II}_p(u, v) = dN_p(u) cdot v$ is a (positive-definite) symmetric billinear form on $S$, also known as the second fundamental form. This is an inner product on the tangent spaces of $S$, but not the standard one (coming from just taking $u cdot v$ in $Bbb R^3$). This is a more extrinsic object, and carries information about the embedding of the surface in $Bbb R^3$ rather than the surface itself.
– Balarka Sen
Dec 27 '18 at 5:03
Thank you for these posts, they were both very enlightening. In your first post you say that the dot product can be interpreted as "the rate of change of the unit normal vector at p, when p is pushed in the direction of u, as perceived by the subspace $<v>$. Thank you a lot for that. Some part of me wants to always consider u and v to be orthogonal vectors in the surface they parameterize, but this is definitely not true, obviously my geometric intuition needs developing.
– Michael Vaughan
Dec 27 '18 at 14:13
1
I'd add that, using the slightly more abstract language of Riemannian geometry, this essentially boils down to the statement that the Levi-Civita connection is torsion-free.
– Or Eisenberg
Dec 27 '18 at 18:50
add a comment |
1
The number $dN_p(u) cdot v$ is roughly the "rate of change of the unit normal vector at $p$, when $p$ is pushed in the direction of $u$, as perceived by the subspace $langle v rangle subset T_p S$ (the dot with $v$ is to be interpreted as projection)". By self-adjointness, this number is symmetric in $u$ and $v$, so rate of change of $N_p$ when $p$ is pushed a little in the direction of $u$, as felt in the direction of $v$ is equal to rate of change of $N_p$ when $p$ is pushed in the direction of $v$, as felt in the direction of $u$. At least this is how I understand the result.
– Balarka Sen
Dec 27 '18 at 4:55
1
More algebraically, this means $Bbb{II}_p(u, v) = dN_p(u) cdot v$ is a (positive-definite) symmetric billinear form on $S$, also known as the second fundamental form. This is an inner product on the tangent spaces of $S$, but not the standard one (coming from just taking $u cdot v$ in $Bbb R^3$). This is a more extrinsic object, and carries information about the embedding of the surface in $Bbb R^3$ rather than the surface itself.
– Balarka Sen
Dec 27 '18 at 5:03
Thank you for these posts, they were both very enlightening. In your first post you say that the dot product can be interpreted as "the rate of change of the unit normal vector at p, when p is pushed in the direction of u, as perceived by the subspace $<v>$. Thank you a lot for that. Some part of me wants to always consider u and v to be orthogonal vectors in the surface they parameterize, but this is definitely not true, obviously my geometric intuition needs developing.
– Michael Vaughan
Dec 27 '18 at 14:13
1
I'd add that, using the slightly more abstract language of Riemannian geometry, this essentially boils down to the statement that the Levi-Civita connection is torsion-free.
– Or Eisenberg
Dec 27 '18 at 18:50
1
1
The number $dN_p(u) cdot v$ is roughly the "rate of change of the unit normal vector at $p$, when $p$ is pushed in the direction of $u$, as perceived by the subspace $langle v rangle subset T_p S$ (the dot with $v$ is to be interpreted as projection)". By self-adjointness, this number is symmetric in $u$ and $v$, so rate of change of $N_p$ when $p$ is pushed a little in the direction of $u$, as felt in the direction of $v$ is equal to rate of change of $N_p$ when $p$ is pushed in the direction of $v$, as felt in the direction of $u$. At least this is how I understand the result.
– Balarka Sen
Dec 27 '18 at 4:55
The number $dN_p(u) cdot v$ is roughly the "rate of change of the unit normal vector at $p$, when $p$ is pushed in the direction of $u$, as perceived by the subspace $langle v rangle subset T_p S$ (the dot with $v$ is to be interpreted as projection)". By self-adjointness, this number is symmetric in $u$ and $v$, so rate of change of $N_p$ when $p$ is pushed a little in the direction of $u$, as felt in the direction of $v$ is equal to rate of change of $N_p$ when $p$ is pushed in the direction of $v$, as felt in the direction of $u$. At least this is how I understand the result.
– Balarka Sen
Dec 27 '18 at 4:55
1
1
More algebraically, this means $Bbb{II}_p(u, v) = dN_p(u) cdot v$ is a (positive-definite) symmetric billinear form on $S$, also known as the second fundamental form. This is an inner product on the tangent spaces of $S$, but not the standard one (coming from just taking $u cdot v$ in $Bbb R^3$). This is a more extrinsic object, and carries information about the embedding of the surface in $Bbb R^3$ rather than the surface itself.
– Balarka Sen
Dec 27 '18 at 5:03
More algebraically, this means $Bbb{II}_p(u, v) = dN_p(u) cdot v$ is a (positive-definite) symmetric billinear form on $S$, also known as the second fundamental form. This is an inner product on the tangent spaces of $S$, but not the standard one (coming from just taking $u cdot v$ in $Bbb R^3$). This is a more extrinsic object, and carries information about the embedding of the surface in $Bbb R^3$ rather than the surface itself.
– Balarka Sen
Dec 27 '18 at 5:03
Thank you for these posts, they were both very enlightening. In your first post you say that the dot product can be interpreted as "the rate of change of the unit normal vector at p, when p is pushed in the direction of u, as perceived by the subspace $<v>$. Thank you a lot for that. Some part of me wants to always consider u and v to be orthogonal vectors in the surface they parameterize, but this is definitely not true, obviously my geometric intuition needs developing.
– Michael Vaughan
Dec 27 '18 at 14:13
Thank you for these posts, they were both very enlightening. In your first post you say that the dot product can be interpreted as "the rate of change of the unit normal vector at p, when p is pushed in the direction of u, as perceived by the subspace $<v>$. Thank you a lot for that. Some part of me wants to always consider u and v to be orthogonal vectors in the surface they parameterize, but this is definitely not true, obviously my geometric intuition needs developing.
– Michael Vaughan
Dec 27 '18 at 14:13
1
1
I'd add that, using the slightly more abstract language of Riemannian geometry, this essentially boils down to the statement that the Levi-Civita connection is torsion-free.
– Or Eisenberg
Dec 27 '18 at 18:50
I'd add that, using the slightly more abstract language of Riemannian geometry, this essentially boils down to the statement that the Levi-Civita connection is torsion-free.
– Or Eisenberg
Dec 27 '18 at 18:50
add a comment |
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1
The number $dN_p(u) cdot v$ is roughly the "rate of change of the unit normal vector at $p$, when $p$ is pushed in the direction of $u$, as perceived by the subspace $langle v rangle subset T_p S$ (the dot with $v$ is to be interpreted as projection)". By self-adjointness, this number is symmetric in $u$ and $v$, so rate of change of $N_p$ when $p$ is pushed a little in the direction of $u$, as felt in the direction of $v$ is equal to rate of change of $N_p$ when $p$ is pushed in the direction of $v$, as felt in the direction of $u$. At least this is how I understand the result.
– Balarka Sen
Dec 27 '18 at 4:55
1
More algebraically, this means $Bbb{II}_p(u, v) = dN_p(u) cdot v$ is a (positive-definite) symmetric billinear form on $S$, also known as the second fundamental form. This is an inner product on the tangent spaces of $S$, but not the standard one (coming from just taking $u cdot v$ in $Bbb R^3$). This is a more extrinsic object, and carries information about the embedding of the surface in $Bbb R^3$ rather than the surface itself.
– Balarka Sen
Dec 27 '18 at 5:03
Thank you for these posts, they were both very enlightening. In your first post you say that the dot product can be interpreted as "the rate of change of the unit normal vector at p, when p is pushed in the direction of u, as perceived by the subspace $<v>$. Thank you a lot for that. Some part of me wants to always consider u and v to be orthogonal vectors in the surface they parameterize, but this is definitely not true, obviously my geometric intuition needs developing.
– Michael Vaughan
Dec 27 '18 at 14:13
1
I'd add that, using the slightly more abstract language of Riemannian geometry, this essentially boils down to the statement that the Levi-Civita connection is torsion-free.
– Or Eisenberg
Dec 27 '18 at 18:50