Mixing
How to express curvature of a level set in terms of derivatives of a function?
$begingroup$
Suppose I have a smooth function $u:mathbb R^ntomathbb R$.
Assume that its gradient doesn't vanish (near any point where we investigate it).
Is there a list of different (intrinsic and extrinsic) curvature quantities of level sets of $u$ in terms of derivatives of $u$?
I have been unable to find such a list.
The level set is a Riemannian manifold and its curvature can be described by various curvature tensors.
It is also a submanifold of the ambient $mathbb R^n$ and the second fundamental form describes its curvature as a submanifold.
These are what I refer to as intrinsic and extrinsic curvature quantities.
Here are two examples of questions that the list should answer.
I am looking for a resource that would contain the answer to these two questions and many others, not just the answer to these two. These example questions give a criterion for what I am looking for, that's all. This question is a reference request.
- If $n=3$, what is the Gaussian curvature of $u^{-1}(u(0))$ at $0$ in terms of derivatives of $u$?
- How to express the mean curvature of the level set in terms of derivatives of $u$ in any dimension?
multivariable-calculus differential-geometry reference-request riemannian-geometry curvature
asked Jul 16 '15 at 11:03
Joonas IlmavirtaJoonas Ilmavirta
20.7k94282
$endgroup$
add a comment |
$begingroup$
Suppose I have a smooth function $u:mathbb R^ntomathbb R$.
Assume that its gradient doesn't vanish (near any point where we investigate it).
Is there a list of different (intrinsic and extrinsic) curvature quantities of level sets of $u$ in terms of derivatives of $u$?
I have been unable to find such a list.
The level set is a Riemannian manifold and its curvature can be described by various curvature tensors.
It is also a submanifold of the ambient $mathbb R^n$ and the second fundamental form describes its curvature as a submanifold.
These are what I refer to as intrinsic and extrinsic curvature quantities.
Here are two examples of questions that the list should answer.
I am looking for a resource that would contain the answer to these two questions and many others, not just the answer to these two. These example questions give a criterion for what I am looking for, that's all. This question is a reference request.
- If $n=3$, what is the Gaussian curvature of $u^{-1}(u(0))$ at $0$ in terms of derivatives of $u$?
- How to express the mean curvature of the level set in terms of derivatives of $u$ in any dimension?
multivariable-calculus differential-geometry reference-request riemannian-geometry curvature
asked Jul 16 '15 at 11:03
Joonas IlmavirtaJoonas Ilmavirta
20.7k94282
$endgroup$
$begingroup$
Have you looked through do Carmo?
$endgroup$
– Neal
Jul 16 '15 at 11:18
$begingroup$
@Neal, I haven't. Based on a quick glance it seems like a probable source, so I will take a look. Thanks for the tip!
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:34
add a comment |
$begingroup$
Suppose I have a smooth function $u:mathbb R^ntomathbb R$.
Assume that its gradient doesn't vanish (near any point where we investigate it).
Is there a list of different (intrinsic and extrinsic) curvature quantities of level sets of $u$ in terms of derivatives of $u$?
I have been unable to find such a list.
The level set is a Riemannian manifold and its curvature can be described by various curvature tensors.
It is also a submanifold of the ambient $mathbb R^n$ and the second fundamental form describes its curvature as a submanifold.
These are what I refer to as intrinsic and extrinsic curvature quantities.
Here are two examples of questions that the list should answer.
I am looking for a resource that would contain the answer to these two questions and many others, not just the answer to these two. These example questions give a criterion for what I am looking for, that's all. This question is a reference request.
- If $n=3$, what is the Gaussian curvature of $u^{-1}(u(0))$ at $0$ in terms of derivatives of $u$?
- How to express the mean curvature of the level set in terms of derivatives of $u$ in any dimension?
multivariable-calculus differential-geometry reference-request riemannian-geometry curvature
asked Jul 16 '15 at 11:03
Joonas IlmavirtaJoonas Ilmavirta
20.7k94282
$endgroup$
Suppose I have a smooth function $u:mathbb R^ntomathbb R$.
Assume that its gradient doesn't vanish (near any point where we investigate it).
Is there a list of different (intrinsic and extrinsic) curvature quantities of level sets of $u$ in terms of derivatives of $u$?
I have been unable to find such a list.
The level set is a Riemannian manifold and its curvature can be described by various curvature tensors.
It is also a submanifold of the ambient $mathbb R^n$ and the second fundamental form describes its curvature as a submanifold.
These are what I refer to as intrinsic and extrinsic curvature quantities.
Here are two examples of questions that the list should answer.
I am looking for a resource that would contain the answer to these two questions and many others, not just the answer to these two. These example questions give a criterion for what I am looking for, that's all. This question is a reference request.
- If $n=3$, what is the Gaussian curvature of $u^{-1}(u(0))$ at $0$ in terms of derivatives of $u$?
- How to express the mean curvature of the level set in terms of derivatives of $u$ in any dimension?
multivariable-calculus differential-geometry reference-request riemannian-geometry curvature
multivariable-calculus differential-geometry reference-request riemannian-geometry curvature
asked Jul 16 '15 at 11:03
Joonas IlmavirtaJoonas Ilmavirta
20.7k94282
asked Jul 16 '15 at 11:03
Joonas IlmavirtaJoonas Ilmavirta
20.7k94282
asked Jul 16 '15 at 11:03
Joonas IlmavirtaJoonas Ilmavirta
20.7k94282
asked Jul 16 '15 at 11:03
Joonas IlmavirtaJoonas Ilmavirta
20.7k94282
asked Jul 16 '15 at 11:03
Joonas IlmavirtaJoonas Ilmavirta
20.7k94282
20.7k94282
$begingroup$
Have you looked through do Carmo?
$endgroup$
– Neal
Jul 16 '15 at 11:18
$begingroup$
@Neal, I haven't. Based on a quick glance it seems like a probable source, so I will take a look. Thanks for the tip!
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:34
add a comment |
$begingroup$
Have you looked through do Carmo?
$endgroup$
– Neal
Jul 16 '15 at 11:18
$begingroup$
@Neal, I haven't. Based on a quick glance it seems like a probable source, so I will take a look. Thanks for the tip!
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:34
$begingroup$
Have you looked through do Carmo?
$endgroup$
– Neal
Jul 16 '15 at 11:18
$begingroup$
Have you looked through do Carmo?
$endgroup$
– Neal
Jul 16 '15 at 11:18
$begingroup$
@Neal, I haven't. Based on a quick glance it seems like a probable source, so I will take a look. Thanks for the tip!
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:34
$begingroup$
@Neal, I haven't. Based on a quick glance it seems like a probable source, so I will take a look. Thanks for the tip!
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:34
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
For the first, you might want to look at a tech report I wrote several years back:
http://cs.brown.edu/people/jhughes/papers/Hughes-DGO-2003/paper.pdf
For the second, there's a quite general paper (of which why tech report is a distillation in the 3D case, with a correction):
Peter Dombrowski. Krümmungsgrößen gleichungsdefinierter untermannigfaltigkeiten riemmannscher mannigfaltigkeiten. Mathematische Nachrichten, 38(3/4):133–180, 1968.
answered Jul 16 '15 at 12:17
John HughesJohn Hughes
63.9k24191
$endgroup$
$begingroup$
Thanks! I will take a look at the report when I have the time. I actually already knew the answer to the second question (the 1-Laplacian of $u$), but I included it as an example because it is in the spirit of things that I would like to see. (I tried to improve the formatting of the German title. I hope I got it right.)
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:26
$begingroup$
@John Hughes, the link you posted is broken nowadays.
$endgroup$
– stewori
Jan 11 at 16:34
1
$begingroup$
Fixed! (They changed my username a few months back, darn it!)
$endgroup$
– John Hughes
Jan 11 at 17:04
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
For the first, you might want to look at a tech report I wrote several years back:
http://cs.brown.edu/people/jhughes/papers/Hughes-DGO-2003/paper.pdf
For the second, there's a quite general paper (of which why tech report is a distillation in the 3D case, with a correction):
Peter Dombrowski. Krümmungsgrößen gleichungsdefinierter untermannigfaltigkeiten riemmannscher mannigfaltigkeiten. Mathematische Nachrichten, 38(3/4):133–180, 1968.
answered Jul 16 '15 at 12:17
John HughesJohn Hughes
63.9k24191
$endgroup$
$begingroup$
Thanks! I will take a look at the report when I have the time. I actually already knew the answer to the second question (the 1-Laplacian of $u$), but I included it as an example because it is in the spirit of things that I would like to see. (I tried to improve the formatting of the German title. I hope I got it right.)
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:26
$begingroup$
@John Hughes, the link you posted is broken nowadays.
$endgroup$
– stewori
Jan 11 at 16:34
1
$begingroup$
Fixed! (They changed my username a few months back, darn it!)
$endgroup$
– John Hughes
Jan 11 at 17:04
add a comment |
$begingroup$
For the first, you might want to look at a tech report I wrote several years back:
http://cs.brown.edu/people/jhughes/papers/Hughes-DGO-2003/paper.pdf
For the second, there's a quite general paper (of which why tech report is a distillation in the 3D case, with a correction):
Peter Dombrowski. Krümmungsgrößen gleichungsdefinierter untermannigfaltigkeiten riemmannscher mannigfaltigkeiten. Mathematische Nachrichten, 38(3/4):133–180, 1968.
answered Jul 16 '15 at 12:17
John HughesJohn Hughes
63.9k24191
$endgroup$
$begingroup$
Thanks! I will take a look at the report when I have the time. I actually already knew the answer to the second question (the 1-Laplacian of $u$), but I included it as an example because it is in the spirit of things that I would like to see. (I tried to improve the formatting of the German title. I hope I got it right.)
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:26
$begingroup$
@John Hughes, the link you posted is broken nowadays.
$endgroup$
– stewori
Jan 11 at 16:34
1
$begingroup$
Fixed! (They changed my username a few months back, darn it!)
$endgroup$
– John Hughes
Jan 11 at 17:04
add a comment |
$begingroup$
For the first, you might want to look at a tech report I wrote several years back:
http://cs.brown.edu/people/jhughes/papers/Hughes-DGO-2003/paper.pdf
For the second, there's a quite general paper (of which why tech report is a distillation in the 3D case, with a correction):
Peter Dombrowski. Krümmungsgrößen gleichungsdefinierter untermannigfaltigkeiten riemmannscher mannigfaltigkeiten. Mathematische Nachrichten, 38(3/4):133–180, 1968.
answered Jul 16 '15 at 12:17
John HughesJohn Hughes
63.9k24191
$endgroup$
For the first, you might want to look at a tech report I wrote several years back:
http://cs.brown.edu/people/jhughes/papers/Hughes-DGO-2003/paper.pdf
For the second, there's a quite general paper (of which why tech report is a distillation in the 3D case, with a correction):
Peter Dombrowski. Krümmungsgrößen gleichungsdefinierter untermannigfaltigkeiten riemmannscher mannigfaltigkeiten. Mathematische Nachrichten, 38(3/4):133–180, 1968.
answered Jul 16 '15 at 12:17
John HughesJohn Hughes
63.9k24191
edited Jan 11 at 17:04
answered Jul 16 '15 at 12:17
John HughesJohn Hughes
63.9k24191
answered Jul 16 '15 at 12:17
John HughesJohn Hughes
63.9k24191
answered Jul 16 '15 at 12:17
John HughesJohn Hughes
63.9k24191
63.9k24191
$begingroup$
Thanks! I will take a look at the report when I have the time. I actually already knew the answer to the second question (the 1-Laplacian of $u$), but I included it as an example because it is in the spirit of things that I would like to see. (I tried to improve the formatting of the German title. I hope I got it right.)
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:26
$begingroup$
@John Hughes, the link you posted is broken nowadays.
$endgroup$
– stewori
Jan 11 at 16:34
1
$begingroup$
Fixed! (They changed my username a few months back, darn it!)
$endgroup$
– John Hughes
Jan 11 at 17:04
add a comment |
$begingroup$
Thanks! I will take a look at the report when I have the time. I actually already knew the answer to the second question (the 1-Laplacian of $u$), but I included it as an example because it is in the spirit of things that I would like to see. (I tried to improve the formatting of the German title. I hope I got it right.)
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:26
$begingroup$
@John Hughes, the link you posted is broken nowadays.
$endgroup$
– stewori
Jan 11 at 16:34
1
$begingroup$
Fixed! (They changed my username a few months back, darn it!)
$endgroup$
– John Hughes
Jan 11 at 17:04
$begingroup$
Thanks! I will take a look at the report when I have the time. I actually already knew the answer to the second question (the 1-Laplacian of $u$), but I included it as an example because it is in the spirit of things that I would like to see. (I tried to improve the formatting of the German title. I hope I got it right.)
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:26
$begingroup$
Thanks! I will take a look at the report when I have the time. I actually already knew the answer to the second question (the 1-Laplacian of $u$), but I included it as an example because it is in the spirit of things that I would like to see. (I tried to improve the formatting of the German title. I hope I got it right.)
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:26
$begingroup$
@John Hughes, the link you posted is broken nowadays.
$endgroup$
– stewori
Jan 11 at 16:34
$begingroup$
@John Hughes, the link you posted is broken nowadays.
$endgroup$
– stewori
Jan 11 at 16:34
1
1
$begingroup$
Fixed! (They changed my username a few months back, darn it!)
$endgroup$
– John Hughes
Jan 11 at 17:04
$begingroup$
Fixed! (They changed my username a few months back, darn it!)
$endgroup$
– John Hughes
Jan 11 at 17:04
add a comment |
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$begingroup$
Have you looked through do Carmo?
$endgroup$
– Neal
Jul 16 '15 at 11:18
$begingroup$
@Neal, I haven't. Based on a quick glance it seems like a probable source, so I will take a look. Thanks for the tip!
$endgroup$
– Joonas Ilmavirta
Jul 16 '15 at 12:34