Using the Gauss-Bonnet theorem to determine a surface
$begingroup$
I'm using the Gauss-Bonnet theorem on the following exercise and I would like to see if what I've done is correct.
Given a compact, connected surface $M$ with area $A(M)=1984707$ and Gauss curvature $K=frac{-31658} {10^6}$, determine the surface.
So, using Gauss-Bonnet to determine the Euler characteristic $χ(M)$ of $M$, I get:
$K*A(M)=2π*χ(M)$ which of course will gives a non integer characteristic.
I'm assuming it's just the numbers given but I thought to ask.
So, did I use Gauss-Bonnet correctly or this is not how it's done?
Also, if I did everything right, is there a way to determine if this surface is orientable or not with the information that the exercise gives?
Again, I'm sorry if this is trivial
differential-geometry geometric-topology
asked Dec 21 '18 at 15:40
AmontilladoAmontillado
449313
$endgroup$
add a comment |
$begingroup$
I'm using the Gauss-Bonnet theorem on the following exercise and I would like to see if what I've done is correct.
Given a compact, connected surface $M$ with area $A(M)=1984707$ and Gauss curvature $K=frac{-31658} {10^6}$, determine the surface.
So, using Gauss-Bonnet to determine the Euler characteristic $χ(M)$ of $M$, I get:
$K*A(M)=2π*χ(M)$ which of course will gives a non integer characteristic.
I'm assuming it's just the numbers given but I thought to ask.
So, did I use Gauss-Bonnet correctly or this is not how it's done?
Also, if I did everything right, is there a way to determine if this surface is orientable or not with the information that the exercise gives?
Again, I'm sorry if this is trivial
differential-geometry geometric-topology
asked Dec 21 '18 at 15:40
AmontilladoAmontillado
449313
$endgroup$
1
$begingroup$
You're doing it right. I think the spirit of the problem is that this arithmetic is going to give a (very good) approximation to an integer.
$endgroup$
– Ted Shifrin
Dec 21 '18 at 17:08
$begingroup$
It does indeed. Thank you!
$endgroup$
– Amontillado
Dec 21 '18 at 19:03
add a comment |
$begingroup$
I'm using the Gauss-Bonnet theorem on the following exercise and I would like to see if what I've done is correct.
Given a compact, connected surface $M$ with area $A(M)=1984707$ and Gauss curvature $K=frac{-31658} {10^6}$, determine the surface.
So, using Gauss-Bonnet to determine the Euler characteristic $χ(M)$ of $M$, I get:
$K*A(M)=2π*χ(M)$ which of course will gives a non integer characteristic.
I'm assuming it's just the numbers given but I thought to ask.
So, did I use Gauss-Bonnet correctly or this is not how it's done?
Also, if I did everything right, is there a way to determine if this surface is orientable or not with the information that the exercise gives?
Again, I'm sorry if this is trivial
differential-geometry geometric-topology
asked Dec 21 '18 at 15:40
AmontilladoAmontillado
449313
$endgroup$
I'm using the Gauss-Bonnet theorem on the following exercise and I would like to see if what I've done is correct.
Given a compact, connected surface $M$ with area $A(M)=1984707$ and Gauss curvature $K=frac{-31658} {10^6}$, determine the surface.
So, using Gauss-Bonnet to determine the Euler characteristic $χ(M)$ of $M$, I get:
$K*A(M)=2π*χ(M)$ which of course will gives a non integer characteristic.
I'm assuming it's just the numbers given but I thought to ask.
So, did I use Gauss-Bonnet correctly or this is not how it's done?
Also, if I did everything right, is there a way to determine if this surface is orientable or not with the information that the exercise gives?
Again, I'm sorry if this is trivial
differential-geometry geometric-topology
differential-geometry geometric-topology
asked Dec 21 '18 at 15:40
AmontilladoAmontillado
449313
asked Dec 21 '18 at 15:40
AmontilladoAmontillado
449313
asked Dec 21 '18 at 15:40
AmontilladoAmontillado
449313
asked Dec 21 '18 at 15:40
AmontilladoAmontillado
449313
asked Dec 21 '18 at 15:40
AmontilladoAmontillado
449313
449313
1
$begingroup$
You're doing it right. I think the spirit of the problem is that this arithmetic is going to give a (very good) approximation to an integer.
$endgroup$
– Ted Shifrin
Dec 21 '18 at 17:08
$begingroup$
It does indeed. Thank you!
$endgroup$
– Amontillado
Dec 21 '18 at 19:03
add a comment |
1
$begingroup$
You're doing it right. I think the spirit of the problem is that this arithmetic is going to give a (very good) approximation to an integer.
$endgroup$
– Ted Shifrin
Dec 21 '18 at 17:08
$begingroup$
It does indeed. Thank you!
$endgroup$
– Amontillado
Dec 21 '18 at 19:03
1
1
$begingroup$
You're doing it right. I think the spirit of the problem is that this arithmetic is going to give a (very good) approximation to an integer.
$endgroup$
– Ted Shifrin
Dec 21 '18 at 17:08
$begingroup$
You're doing it right. I think the spirit of the problem is that this arithmetic is going to give a (very good) approximation to an integer.
$endgroup$
– Ted Shifrin
Dec 21 '18 at 17:08
$begingroup$
It does indeed. Thank you!
$endgroup$
– Amontillado
Dec 21 '18 at 19:03
$begingroup$
It does indeed. Thank you!
$endgroup$
– Amontillado
Dec 21 '18 at 19:03
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3048622%2fusing-the-gauss-bonnet-theorem-to-determine-a-surface%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3048622%2fusing-the-gauss-bonnet-theorem-to-determine-a-surface%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
You're doing it right. I think the spirit of the problem is that this arithmetic is going to give a (very good) approximation to an integer.
$endgroup$
– Ted Shifrin
Dec 21 '18 at 17:08
$begingroup$
It does indeed. Thank you!
$endgroup$
– Amontillado
Dec 21 '18 at 19:03