Mixing
Why would the following define a 1-1 tensor?
$begingroup$
Let $g$ be a Riemannian metric on a manifold $M$, and let $mu_g$ be the volume element, i.e. for any $x in M, v,w in T_x M$, we have
$$mu_g(v,w)=pmsqrt{detbegin{bmatrix}
g(x)(v,v) & g(x)(v,w)\
g(x)(w,v) & g(x)(w,w)
end{bmatrix}}$$
Let $g^{-1}$ be the inverse metric and it was claim in a text that the map $g^{-1}mu_g$ is a 1-1 tensor. I think that the text regard $v mapsto mu_g(v,bullet)$ as a map from $T_x M$ to $T_xM^*$ for any fixed $x in M$. But I can't see why the map $g^{-1}mu_g$ is linear as $mu_g$ may not be linear.
Remark: The above map is mentioned in page 21, Theorem 1.3.4 of the book Teichmuller Theory in Riemannian Geometry.
differential-geometry riemannian-geometry tensors
asked Jan 22 at 4:05
JerryJerry
457313
$endgroup$
add a comment |
$begingroup$
Let $g$ be a Riemannian metric on a manifold $M$, and let $mu_g$ be the volume element, i.e. for any $x in M, v,w in T_x M$, we have
$$mu_g(v,w)=pmsqrt{detbegin{bmatrix}
g(x)(v,v) & g(x)(v,w)\
g(x)(w,v) & g(x)(w,w)
end{bmatrix}}$$
Let $g^{-1}$ be the inverse metric and it was claim in a text that the map $g^{-1}mu_g$ is a 1-1 tensor. I think that the text regard $v mapsto mu_g(v,bullet)$ as a map from $T_x M$ to $T_xM^*$ for any fixed $x in M$. But I can't see why the map $g^{-1}mu_g$ is linear as $mu_g$ may not be linear.
Remark: The above map is mentioned in page 21, Theorem 1.3.4 of the book Teichmuller Theory in Riemannian Geometry.
differential-geometry riemannian-geometry tensors
asked Jan 22 at 4:05
JerryJerry
457313
$endgroup$
add a comment |
$begingroup$
Let $g$ be a Riemannian metric on a manifold $M$, and let $mu_g$ be the volume element, i.e. for any $x in M, v,w in T_x M$, we have
$$mu_g(v,w)=pmsqrt{detbegin{bmatrix}
g(x)(v,v) & g(x)(v,w)\
g(x)(w,v) & g(x)(w,w)
end{bmatrix}}$$
Let $g^{-1}$ be the inverse metric and it was claim in a text that the map $g^{-1}mu_g$ is a 1-1 tensor. I think that the text regard $v mapsto mu_g(v,bullet)$ as a map from $T_x M$ to $T_xM^*$ for any fixed $x in M$. But I can't see why the map $g^{-1}mu_g$ is linear as $mu_g$ may not be linear.
Remark: The above map is mentioned in page 21, Theorem 1.3.4 of the book Teichmuller Theory in Riemannian Geometry.
differential-geometry riemannian-geometry tensors
asked Jan 22 at 4:05
JerryJerry
457313
$endgroup$
Let $g$ be a Riemannian metric on a manifold $M$, and let $mu_g$ be the volume element, i.e. for any $x in M, v,w in T_x M$, we have
$$mu_g(v,w)=pmsqrt{detbegin{bmatrix}
g(x)(v,v) & g(x)(v,w)\
g(x)(w,v) & g(x)(w,w)
end{bmatrix}}$$
Let $g^{-1}$ be the inverse metric and it was claim in a text that the map $g^{-1}mu_g$ is a 1-1 tensor. I think that the text regard $v mapsto mu_g(v,bullet)$ as a map from $T_x M$ to $T_xM^*$ for any fixed $x in M$. But I can't see why the map $g^{-1}mu_g$ is linear as $mu_g$ may not be linear.
Remark: The above map is mentioned in page 21, Theorem 1.3.4 of the book Teichmuller Theory in Riemannian Geometry.
differential-geometry riemannian-geometry tensors
differential-geometry riemannian-geometry tensors
asked Jan 22 at 4:05
JerryJerry
457313
asked Jan 22 at 4:05
JerryJerry
457313
asked Jan 22 at 4:05
JerryJerry
457313
asked Jan 22 at 4:05
JerryJerry
457313
asked Jan 22 at 4:05
JerryJerry
457313
457313
add a comment |
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%2f3082733%2fwhy-would-the-following-define-a-1-1-tensor%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%2f3082733%2fwhy-would-the-following-define-a-1-1-tensor%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
