How is the push forward on the tangent spaces defined? The 2019 Stack Overflow Developer Survey Results Are InUnderstanding the Definition of a Differential Form of Degree $k$Intuition about pullbacks in differential geometryIf a smooth map between manifolds is injective, is the induced map on the tangent spaces injective too?What is the pushforward of a function (not a vector)Vector Bundle Structure on $sqcup_pin Mmathcal L(T_pM, T_f(p)N)$.On a characterization of tangent spaces of submanifoldsCan every linear map between tangent spaces be realized as a differential of a map?Non-Injective differentiable map between manifoldsTangent space to a product manifold using curvesQuestion regarding tangent spaces of a Euclidean space
Earliest use of the term "Galois extension"?
Button changing it's text & action. Good or terrible?
Loose spokes after only a few rides
Should I use my personal e-mail address, or my workplace one, when registering to external websites for work purposes?
Apparent duplicates between Haynes service instructions and MOT
Why was M87 targetted for the Event Horizon Telescope instead of Sagittarius A*?
What does Linus Torvalds mean when he says that Git "never ever" tracks a file?
Have you ever entered Singapore using a different passport or name?
How to manage monthly salary
Multiply Two Integer Polynomials
What to do when moving next to a bird sanctuary with a loosely-domesticated cat?
Identify This Plant (Flower)
How to support a colleague who finds meetings extremely tiring?
Falsification in Math vs Science
How technical should a Scrum Master be to effectively remove impediments?
"as much details as you can remember"
Which Sci-Fi work first showed weapon of galactic-scale mass destruction?
Why did Acorn's A3000 have red function keys?
Is this app Icon Browser Safe/Legit?
Origin of "cooter" meaning "vagina"
Is there any way to tell whether the shot is going to hit you or not?
Did Scotland spend $250,000 for the slogan "Welcome to Scotland"?
Do these rules for Critical Successes and Critical Failures seem Fair?
Write faster on AT24C32
How is the push forward on the tangent spaces defined?
The 2019 Stack Overflow Developer Survey Results Are InUnderstanding the Definition of a Differential Form of Degree $k$Intuition about pullbacks in differential geometryIf a smooth map between manifolds is injective, is the induced map on the tangent spaces injective too?What is the pushforward of a function (not a vector)Vector Bundle Structure on $sqcup_pin Mmathcal L(T_pM, T_f(p)N)$.On a characterization of tangent spaces of submanifoldsCan every linear map between tangent spaces be realized as a differential of a map?Non-Injective differentiable map between manifoldsTangent space to a product manifold using curvesQuestion regarding tangent spaces of a Euclidean space
$begingroup$
Let $F:M rightarrow N$ be a $C^infty$ map between manifolds. Then we have the pullback $F^*:C^infty(N) rightarrow C^infty(M)$. From this we can get a map from
$J_N, F(p) / J_N, F(p)^2 $ to $J_M, p / J_M,p^2$, where $J_M,p$ is the ideal of functions $f$ in $C^infty(M)$ such that $f(p) = 0$. This then gives a map $F^*: T^*_F(p)N rightarrow T^*_p M$ of cotangent spaces (because $T^*_p M$ is isomorphic to $J_M, p / J_M,p^2$ etc). In the notes I am reading then states:
Dually, there is a map $F_*: T_p M rightarrow T_F(p)N$ of tangent spaces.
I am wondering could someone explain to me how this map $F_*$ is defined?
(since there is no explanation in this notes...)
Thank you.
PS here $T_pM$ is the collection of linear maps $X : C^infty(M) rightarrow mathbbR $ such that
$$
X(f_1 f_2) = f_1(p) X f_2 + f_2(p) X f_1.
$$
linear-algebra differential-geometry smooth-manifolds
asked Mar 30 at 16:50
Takeshi GoudaTakeshi Gouda
404
$endgroup$
add a comment |
$begingroup$
Let $F:M rightarrow N$ be a $C^infty$ map between manifolds. Then we have the pullback $F^*:C^infty(N) rightarrow C^infty(M)$. From this we can get a map from
$J_N, F(p) / J_N, F(p)^2 $ to $J_M, p / J_M,p^2$, where $J_M,p$ is the ideal of functions $f$ in $C^infty(M)$ such that $f(p) = 0$. This then gives a map $F^*: T^*_F(p)N rightarrow T^*_p M$ of cotangent spaces (because $T^*_p M$ is isomorphic to $J_M, p / J_M,p^2$ etc). In the notes I am reading then states:
Dually, there is a map $F_*: T_p M rightarrow T_F(p)N$ of tangent spaces.
I am wondering could someone explain to me how this map $F_*$ is defined?
(since there is no explanation in this notes...)
Thank you.
PS here $T_pM$ is the collection of linear maps $X : C^infty(M) rightarrow mathbbR $ such that
$$
X(f_1 f_2) = f_1(p) X f_2 + f_2(p) X f_1.
$$
linear-algebra differential-geometry smooth-manifolds
asked Mar 30 at 16:50
Takeshi GoudaTakeshi Gouda
404
$endgroup$
add a comment |
$begingroup$
Let $F:M rightarrow N$ be a $C^infty$ map between manifolds. Then we have the pullback $F^*:C^infty(N) rightarrow C^infty(M)$. From this we can get a map from
$J_N, F(p) / J_N, F(p)^2 $ to $J_M, p / J_M,p^2$, where $J_M,p$ is the ideal of functions $f$ in $C^infty(M)$ such that $f(p) = 0$. This then gives a map $F^*: T^*_F(p)N rightarrow T^*_p M$ of cotangent spaces (because $T^*_p M$ is isomorphic to $J_M, p / J_M,p^2$ etc). In the notes I am reading then states:
Dually, there is a map $F_*: T_p M rightarrow T_F(p)N$ of tangent spaces.
I am wondering could someone explain to me how this map $F_*$ is defined?
(since there is no explanation in this notes...)
Thank you.
PS here $T_pM$ is the collection of linear maps $X : C^infty(M) rightarrow mathbbR $ such that
$$
X(f_1 f_2) = f_1(p) X f_2 + f_2(p) X f_1.
$$
linear-algebra differential-geometry smooth-manifolds
asked Mar 30 at 16:50
Takeshi GoudaTakeshi Gouda
404
$endgroup$
Let $F:M rightarrow N$ be a $C^infty$ map between manifolds. Then we have the pullback $F^*:C^infty(N) rightarrow C^infty(M)$. From this we can get a map from
$J_N, F(p) / J_N, F(p)^2 $ to $J_M, p / J_M,p^2$, where $J_M,p$ is the ideal of functions $f$ in $C^infty(M)$ such that $f(p) = 0$. This then gives a map $F^*: T^*_F(p)N rightarrow T^*_p M$ of cotangent spaces (because $T^*_p M$ is isomorphic to $J_M, p / J_M,p^2$ etc). In the notes I am reading then states:
Dually, there is a map $F_*: T_p M rightarrow T_F(p)N$ of tangent spaces.
I am wondering could someone explain to me how this map $F_*$ is defined?
(since there is no explanation in this notes...)
Thank you.
PS here $T_pM$ is the collection of linear maps $X : C^infty(M) rightarrow mathbbR $ such that
$$
X(f_1 f_2) = f_1(p) X f_2 + f_2(p) X f_1.
$$
linear-algebra differential-geometry smooth-manifolds
linear-algebra differential-geometry smooth-manifolds
asked Mar 30 at 16:50
Takeshi GoudaTakeshi Gouda
404
asked Mar 30 at 16:50
Takeshi GoudaTakeshi Gouda
404
asked Mar 30 at 16:50
Takeshi GoudaTakeshi Gouda
404
asked Mar 30 at 16:50
Takeshi GoudaTakeshi Gouda
404
asked Mar 30 at 16:50
Takeshi GoudaTakeshi Gouda
404
404
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $X : C^infty(M) rightarrow mathbbR $ such that
$$
X(f_1 f_2) = f_1(p) X f_2 + f_2(p) X f_1,$$
then $F_*:T_pM rightarrow T_f(p)M$ is defined as $(F_*X)(f) = X(fcirc F).$
For more information, I recommend Lee's great book "Introduction to Smooth Manifolds", chapter 3.
answered Mar 30 at 19:45
Flying DogfishFlying Dogfish
30312
$endgroup$
add a comment |
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%2f3168515%2fhow-is-the-push-forward-on-the-tangent-spaces-defined%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Let $X : C^infty(M) rightarrow mathbbR $ such that
$$
X(f_1 f_2) = f_1(p) X f_2 + f_2(p) X f_1,$$
then $F_*:T_pM rightarrow T_f(p)M$ is defined as $(F_*X)(f) = X(fcirc F).$
For more information, I recommend Lee's great book "Introduction to Smooth Manifolds", chapter 3.
answered Mar 30 at 19:45
Flying DogfishFlying Dogfish
30312
$endgroup$
add a comment |
$begingroup$
Let $X : C^infty(M) rightarrow mathbbR $ such that
$$
X(f_1 f_2) = f_1(p) X f_2 + f_2(p) X f_1,$$
then $F_*:T_pM rightarrow T_f(p)M$ is defined as $(F_*X)(f) = X(fcirc F).$
For more information, I recommend Lee's great book "Introduction to Smooth Manifolds", chapter 3.
answered Mar 30 at 19:45
Flying DogfishFlying Dogfish
30312
$endgroup$
add a comment |
$begingroup$
Let $X : C^infty(M) rightarrow mathbbR $ such that
$$
X(f_1 f_2) = f_1(p) X f_2 + f_2(p) X f_1,$$
then $F_*:T_pM rightarrow T_f(p)M$ is defined as $(F_*X)(f) = X(fcirc F).$
For more information, I recommend Lee's great book "Introduction to Smooth Manifolds", chapter 3.
answered Mar 30 at 19:45
Flying DogfishFlying Dogfish
30312
$endgroup$
Let $X : C^infty(M) rightarrow mathbbR $ such that
$$
X(f_1 f_2) = f_1(p) X f_2 + f_2(p) X f_1,$$
then $F_*:T_pM rightarrow T_f(p)M$ is defined as $(F_*X)(f) = X(fcirc F).$
For more information, I recommend Lee's great book "Introduction to Smooth Manifolds", chapter 3.
answered Mar 30 at 19:45
Flying DogfishFlying Dogfish
30312
answered Mar 30 at 19:45
Flying DogfishFlying Dogfish
30312
answered Mar 30 at 19:45
Flying DogfishFlying Dogfish
30312
answered Mar 30 at 19:45
Flying DogfishFlying Dogfish
30312
30312
add a comment |
add a comment |
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%2f3168515%2fhow-is-the-push-forward-on-the-tangent-spaces-defined%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
