Suppose $ dim(M)<dim(N)$, can there be a diffeomorphism from manifold $M$ to a submanifold of the manifold...
$begingroup$
Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine $f$ going from dimension $n$ to dimension $k$. If $n < k$ then $Df_x$ could never be surjective; and if $n > k$ then $Df_x$ could never be injective. In both cases, therefore, $Df_x$ fails to be a bijection.
The definition is-
$f : M → N$ is called a diffeomorphism if, in coordinate charts, it satisfies the definition above. More precisely: Pick any cover of $M$ by compatible coordinate charts and do the same for $N$. Let $φ$ and $ψ$ be charts on, respectively, $M$ and $N$, with $U$ and $V$ as, respectively, the images of $φ$ and $ψ$. The map $ψfφ^{-1} : U → V$ is then a diffeomorphism as in the definition above, whenever $f(φ^{-1}(U)) ⊂ ψ^{-1}(V).$
$ underline{text{My question:}}$
$(1)$ Can there exists a diffeomorphism between two manifolds of different dimensions?
$(2)$ Suppose $dim(M)<dim(N)$, can there be a diffeomorphism from manifold $M$ to a submanifold of the manifold $N$?
Help me
differential-geometry diffeomorphism
edited Jan 4 at 11:39
Davide Giraudo
126k16150261
asked Jan 4 at 10:49
M. A. SARKARM. A. SARKAR
2,1911619
$endgroup$
add a comment |
$begingroup$
Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine $f$ going from dimension $n$ to dimension $k$. If $n < k$ then $Df_x$ could never be surjective; and if $n > k$ then $Df_x$ could never be injective. In both cases, therefore, $Df_x$ fails to be a bijection.
The definition is-
$f : M → N$ is called a diffeomorphism if, in coordinate charts, it satisfies the definition above. More precisely: Pick any cover of $M$ by compatible coordinate charts and do the same for $N$. Let $φ$ and $ψ$ be charts on, respectively, $M$ and $N$, with $U$ and $V$ as, respectively, the images of $φ$ and $ψ$. The map $ψfφ^{-1} : U → V$ is then a diffeomorphism as in the definition above, whenever $f(φ^{-1}(U)) ⊂ ψ^{-1}(V).$
$ underline{text{My question:}}$
$(1)$ Can there exists a diffeomorphism between two manifolds of different dimensions?
$(2)$ Suppose $dim(M)<dim(N)$, can there be a diffeomorphism from manifold $M$ to a submanifold of the manifold $N$?
Help me
differential-geometry diffeomorphism
edited Jan 4 at 11:39
Davide Giraudo
126k16150261
asked Jan 4 at 10:49
M. A. SARKARM. A. SARKAR
2,1911619
$endgroup$
2
$begingroup$
1: Are you not satisfied with the explanation in your first paragraph as to why you can't have diffeomorphisms between manifolds of different dimensions? 2: Consider $M = mathbb R$, $N = mathbb R^2$ and $f : M to N$ sending $f(x) = (x,0)$.
$endgroup$
– Kenny Wong
Jan 4 at 11:18
$begingroup$
@KennyWong, Yes I am satisfied but I still has the confusion regarding my question $(2)$. The map $f(x)=(x,0)$ is a diffeomorphism. So here be a diffeomorphism from manifold $M$ to a submanifold of the manifold $N$. Am I right?
$endgroup$
– M. A. SARKAR
Jan 4 at 11:31
1
$begingroup$
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
$endgroup$
– 0x539
Jan 4 at 12:00
add a comment |
$begingroup$
Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine $f$ going from dimension $n$ to dimension $k$. If $n < k$ then $Df_x$ could never be surjective; and if $n > k$ then $Df_x$ could never be injective. In both cases, therefore, $Df_x$ fails to be a bijection.
The definition is-
$f : M → N$ is called a diffeomorphism if, in coordinate charts, it satisfies the definition above. More precisely: Pick any cover of $M$ by compatible coordinate charts and do the same for $N$. Let $φ$ and $ψ$ be charts on, respectively, $M$ and $N$, with $U$ and $V$ as, respectively, the images of $φ$ and $ψ$. The map $ψfφ^{-1} : U → V$ is then a diffeomorphism as in the definition above, whenever $f(φ^{-1}(U)) ⊂ ψ^{-1}(V).$
$ underline{text{My question:}}$
$(1)$ Can there exists a diffeomorphism between two manifolds of different dimensions?
$(2)$ Suppose $dim(M)<dim(N)$, can there be a diffeomorphism from manifold $M$ to a submanifold of the manifold $N$?
Help me
differential-geometry diffeomorphism
edited Jan 4 at 11:39
Davide Giraudo
126k16150261
asked Jan 4 at 10:49
M. A. SARKARM. A. SARKAR
2,1911619
$endgroup$
Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine $f$ going from dimension $n$ to dimension $k$. If $n < k$ then $Df_x$ could never be surjective; and if $n > k$ then $Df_x$ could never be injective. In both cases, therefore, $Df_x$ fails to be a bijection.
The definition is-
$f : M → N$ is called a diffeomorphism if, in coordinate charts, it satisfies the definition above. More precisely: Pick any cover of $M$ by compatible coordinate charts and do the same for $N$. Let $φ$ and $ψ$ be charts on, respectively, $M$ and $N$, with $U$ and $V$ as, respectively, the images of $φ$ and $ψ$. The map $ψfφ^{-1} : U → V$ is then a diffeomorphism as in the definition above, whenever $f(φ^{-1}(U)) ⊂ ψ^{-1}(V).$
$ underline{text{My question:}}$
$(1)$ Can there exists a diffeomorphism between two manifolds of different dimensions?
$(2)$ Suppose $dim(M)<dim(N)$, can there be a diffeomorphism from manifold $M$ to a submanifold of the manifold $N$?
Help me
differential-geometry diffeomorphism
differential-geometry diffeomorphism
edited Jan 4 at 11:39
Davide Giraudo
126k16150261
asked Jan 4 at 10:49
M. A. SARKARM. A. SARKAR
2,1911619
edited Jan 4 at 11:39
Davide Giraudo
126k16150261
asked Jan 4 at 10:49
M. A. SARKARM. A. SARKAR
2,1911619
edited Jan 4 at 11:39
Davide Giraudo
126k16150261
edited Jan 4 at 11:39
Davide Giraudo
126k16150261
edited Jan 4 at 11:39
Davide Giraudo
126k16150261
126k16150261
asked Jan 4 at 10:49
M. A. SARKARM. A. SARKAR
2,1911619
asked Jan 4 at 10:49
M. A. SARKARM. A. SARKAR
2,1911619
asked Jan 4 at 10:49
M. A. SARKARM. A. SARKAR
2,1911619
2,1911619
2
$begingroup$
1: Are you not satisfied with the explanation in your first paragraph as to why you can't have diffeomorphisms between manifolds of different dimensions? 2: Consider $M = mathbb R$, $N = mathbb R^2$ and $f : M to N$ sending $f(x) = (x,0)$.
$endgroup$
– Kenny Wong
Jan 4 at 11:18
$begingroup$
@KennyWong, Yes I am satisfied but I still has the confusion regarding my question $(2)$. The map $f(x)=(x,0)$ is a diffeomorphism. So here be a diffeomorphism from manifold $M$ to a submanifold of the manifold $N$. Am I right?
$endgroup$
– M. A. SARKAR
Jan 4 at 11:31
1
$begingroup$
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
$endgroup$
– 0x539
Jan 4 at 12:00
add a comment |
2
$begingroup$
1: Are you not satisfied with the explanation in your first paragraph as to why you can't have diffeomorphisms between manifolds of different dimensions? 2: Consider $M = mathbb R$, $N = mathbb R^2$ and $f : M to N$ sending $f(x) = (x,0)$.
$endgroup$
– Kenny Wong
Jan 4 at 11:18
$begingroup$
@KennyWong, Yes I am satisfied but I still has the confusion regarding my question $(2)$. The map $f(x)=(x,0)$ is a diffeomorphism. So here be a diffeomorphism from manifold $M$ to a submanifold of the manifold $N$. Am I right?
$endgroup$
– M. A. SARKAR
Jan 4 at 11:31
1
$begingroup$
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
$endgroup$
– 0x539
Jan 4 at 12:00
2
2
$begingroup$
1: Are you not satisfied with the explanation in your first paragraph as to why you can't have diffeomorphisms between manifolds of different dimensions? 2: Consider $M = mathbb R$, $N = mathbb R^2$ and $f : M to N$ sending $f(x) = (x,0)$.
$endgroup$
– Kenny Wong
Jan 4 at 11:18
$begingroup$
1: Are you not satisfied with the explanation in your first paragraph as to why you can't have diffeomorphisms between manifolds of different dimensions? 2: Consider $M = mathbb R$, $N = mathbb R^2$ and $f : M to N$ sending $f(x) = (x,0)$.
$endgroup$
– Kenny Wong
Jan 4 at 11:18
$begingroup$
@KennyWong, Yes I am satisfied but I still has the confusion regarding my question $(2)$. The map $f(x)=(x,0)$ is a diffeomorphism. So here be a diffeomorphism from manifold $M$ to a submanifold of the manifold $N$. Am I right?
$endgroup$
– M. A. SARKAR
Jan 4 at 11:31
$begingroup$
@KennyWong, Yes I am satisfied but I still has the confusion regarding my question $(2)$. The map $f(x)=(x,0)$ is a diffeomorphism. So here be a diffeomorphism from manifold $M$ to a submanifold of the manifold $N$. Am I right?
$endgroup$
– M. A. SARKAR
Jan 4 at 11:31
1
1
$begingroup$
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
$endgroup$
– 0x539
Jan 4 at 12:00
$begingroup$
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
$endgroup$
– 0x539
Jan 4 at 12:00
add a comment |
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2
$begingroup$
1: Are you not satisfied with the explanation in your first paragraph as to why you can't have diffeomorphisms between manifolds of different dimensions? 2: Consider $M = mathbb R$, $N = mathbb R^2$ and $f : M to N$ sending $f(x) = (x,0)$.
$endgroup$
– Kenny Wong
Jan 4 at 11:18
$begingroup$
@KennyWong, Yes I am satisfied but I still has the confusion regarding my question $(2)$. The map $f(x)=(x,0)$ is a diffeomorphism. So here be a diffeomorphism from manifold $M$ to a submanifold of the manifold $N$. Am I right?
$endgroup$
– M. A. SARKAR
Jan 4 at 11:31
1
$begingroup$
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
$endgroup$
– 0x539
Jan 4 at 12:00