Suppose $ dim(M)<dim(N)$, can there be a diffeomorphism from manifold $M$ to a submanifold of the manifold...
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 13 hours ago
Davide Giraudo
125k16150260
asked 14 hours ago
M. A. SARKAR
2,1901619
add a comment |
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 13 hours ago
Davide Giraudo
125k16150260
asked 14 hours ago
M. A. SARKAR
2,1901619
1
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 $f(x) = (x,0)$.
– Kenny Wong
13 hours ago
@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?
– M. A. SARKAR
13 hours ago
1
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
– 0x539
13 hours ago
add a comment |
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 13 hours ago
Davide Giraudo
125k16150260
asked 14 hours ago
M. A. SARKAR
2,1901619
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 13 hours ago
Davide Giraudo
125k16150260
asked 14 hours ago
M. A. SARKAR
2,1901619
edited 13 hours ago
Davide Giraudo
125k16150260
asked 14 hours ago
M. A. SARKAR
2,1901619
edited 13 hours ago
Davide Giraudo
125k16150260
edited 13 hours ago
Davide Giraudo
125k16150260
edited 13 hours ago
Davide Giraudo
125k16150260
125k16150260
asked 14 hours ago
M. A. SARKAR
2,1901619
asked 14 hours ago
M. A. SARKAR
2,1901619
asked 14 hours ago
M. A. SARKAR
2,1901619
2,1901619
1
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 $f(x) = (x,0)$.
– Kenny Wong
13 hours ago
@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?
– M. A. SARKAR
13 hours ago
1
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
– 0x539
13 hours ago
add a comment |
1
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 $f(x) = (x,0)$.
– Kenny Wong
13 hours ago
@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?
– M. A. SARKAR
13 hours ago
1
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
– 0x539
13 hours ago
1
1
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 $f(x) = (x,0)$.
– Kenny Wong
13 hours ago
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 $f(x) = (x,0)$.
– Kenny Wong
13 hours ago
@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?
– M. A. SARKAR
13 hours ago
@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?
– M. A. SARKAR
13 hours ago
1
1
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
– 0x539
13 hours ago
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
– 0x539
13 hours ago
add a comment |
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1
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 $f(x) = (x,0)$.
– Kenny Wong
13 hours ago
@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?
– M. A. SARKAR
13 hours ago
1
@M.A.SARKAR Yes you're right (assuming you have $f: mathbb{R} to mathbb{R}^2$ or something similar).
– 0x539
13 hours ago