Mixing
$1$-parameter group terminology problem.
$begingroup$
I'm reading Kobayashi's book Transformation Groups in Differential Geometry, and I'm a bit confused about the terminology that he is using at the page 3. Here is the section that I don't get:
I know that if I have an vector field $X$ on $M$ then I get an $1$-parameter group $varphi_t$ of diffeo's of $M$ by solving a differential equation, namely $frac{dvarphi_t}{dt}=X_{varphi_t}$ plus an initial condition. But $varphi_t$ is not necessarily defined for all $tin mathbb{R}.$
In the statement of the theorem he says "global $1$-parameter groups". My first question is, those 1-parameter groups are defined for all $tin mathbb{R}?$
In the proof he says "1-parameter local group of local transformation of $M$". This I don't get it at all... what is the meaning of those two "local" words?
And as an extra question,on the second raw of the proof, how is $tilde{G}$ define? Can some one point me to a reference?
manifolds lie-groups lie-algebras smooth-manifolds vector-fields
edited Feb 1 at 14:22
J. W. Tanner
4,7121420
asked Feb 1 at 13:22
Hurjui IonutHurjui Ionut
501412
$endgroup$
add a comment |
$begingroup$
I'm reading Kobayashi's book Transformation Groups in Differential Geometry, and I'm a bit confused about the terminology that he is using at the page 3. Here is the section that I don't get:
I know that if I have an vector field $X$ on $M$ then I get an $1$-parameter group $varphi_t$ of diffeo's of $M$ by solving a differential equation, namely $frac{dvarphi_t}{dt}=X_{varphi_t}$ plus an initial condition. But $varphi_t$ is not necessarily defined for all $tin mathbb{R}.$
In the statement of the theorem he says "global $1$-parameter groups". My first question is, those 1-parameter groups are defined for all $tin mathbb{R}?$
In the proof he says "1-parameter local group of local transformation of $M$". This I don't get it at all... what is the meaning of those two "local" words?
And as an extra question,on the second raw of the proof, how is $tilde{G}$ define? Can some one point me to a reference?
manifolds lie-groups lie-algebras smooth-manifolds vector-fields
edited Feb 1 at 14:22
J. W. Tanner
4,7121420
asked Feb 1 at 13:22
Hurjui IonutHurjui Ionut
501412
$endgroup$
add a comment |
$begingroup$
I'm reading Kobayashi's book Transformation Groups in Differential Geometry, and I'm a bit confused about the terminology that he is using at the page 3. Here is the section that I don't get:
I know that if I have an vector field $X$ on $M$ then I get an $1$-parameter group $varphi_t$ of diffeo's of $M$ by solving a differential equation, namely $frac{dvarphi_t}{dt}=X_{varphi_t}$ plus an initial condition. But $varphi_t$ is not necessarily defined for all $tin mathbb{R}.$
In the statement of the theorem he says "global $1$-parameter groups". My first question is, those 1-parameter groups are defined for all $tin mathbb{R}?$
In the proof he says "1-parameter local group of local transformation of $M$". This I don't get it at all... what is the meaning of those two "local" words?
And as an extra question,on the second raw of the proof, how is $tilde{G}$ define? Can some one point me to a reference?
manifolds lie-groups lie-algebras smooth-manifolds vector-fields
edited Feb 1 at 14:22
J. W. Tanner
4,7121420
asked Feb 1 at 13:22
Hurjui IonutHurjui Ionut
501412
$endgroup$
I'm reading Kobayashi's book Transformation Groups in Differential Geometry, and I'm a bit confused about the terminology that he is using at the page 3. Here is the section that I don't get:
I know that if I have an vector field $X$ on $M$ then I get an $1$-parameter group $varphi_t$ of diffeo's of $M$ by solving a differential equation, namely $frac{dvarphi_t}{dt}=X_{varphi_t}$ plus an initial condition. But $varphi_t$ is not necessarily defined for all $tin mathbb{R}.$
In the statement of the theorem he says "global $1$-parameter groups". My first question is, those 1-parameter groups are defined for all $tin mathbb{R}?$
In the proof he says "1-parameter local group of local transformation of $M$". This I don't get it at all... what is the meaning of those two "local" words?
And as an extra question,on the second raw of the proof, how is $tilde{G}$ define? Can some one point me to a reference?
manifolds lie-groups lie-algebras smooth-manifolds vector-fields
manifolds lie-groups lie-algebras smooth-manifolds vector-fields
edited Feb 1 at 14:22
J. W. Tanner
4,7121420
asked Feb 1 at 13:22
Hurjui IonutHurjui Ionut
501412
edited Feb 1 at 14:22
J. W. Tanner
4,7121420
asked Feb 1 at 13:22
Hurjui IonutHurjui Ionut
501412
edited Feb 1 at 14:22
J. W. Tanner
4,7121420
edited Feb 1 at 14:22
J. W. Tanner
4,7121420
edited Feb 1 at 14:22
J. W. Tanner
4,7121420
4,7121420
asked Feb 1 at 13:22
Hurjui IonutHurjui Ionut
501412
asked Feb 1 at 13:22
Hurjui IonutHurjui Ionut
501412
asked Feb 1 at 13:22
Hurjui IonutHurjui Ionut
501412
501412
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Loosely speaking, "global" and "local" in this context simply record whether the expression $phi_t(x)$ in question is defined for all $x in M$ and all $t in mathbb R$, or instead just for $x,t$ in some particular open set (I'm using an augmented notation $phi_t(x)$ where $x$ is put in as a function parameter to represent the initial condition).
For example, the hypothesis of the theorem refers to vector fields $X$ which generate "global 1-parameter groups $phi_t = exp tX$ of $M$", meaning that the solution curves of the differential equation $frac{dphi_t}{dt} = X_{phi_t}$ are defined for all initial conditions $x in M$ and for all $t in mathbb R$.
On the other hand, in general (as I'm sure you know) a vector field $X$ does not generate global 1-parameter groups. Instead, the actual conclusions of the existence/uniqueness theorem for solutions are often recorded using "local" terminology. To say that $X$ generates "local 1-parameter groups of local transformations of $M$" is simply to write down the conclusions of the existence/uniqueness theorem, or perhaps to write them down in a particular manner which emphasizes the "localness". Here's one such way to write the conclusions:
For each $x in M$ there exists an open neighborhood $U subset M$ of $x$ and an open interval $(-epsilon,+epsilon)$ with the following properties:
For each $y in U$ the solution curve $t mapsto phi_{t}(y)$ exists for all $t in (-epsilon,+epsilon)$, and it satisfies the condition that $frac{d}{dt}(phi_t(y)) = X_{phi_t(y)}$.
The map $U times (-epsilon,+epsilon) mapsto M$ defined by $(y,t) mapsto phi_t(y)$ is smooth.
For each fixed $t in (-epsilon,+epsilon)$ the map $U to M$ defined by $y mapsto phi_t(y)$ is a diffeomorphism from $U$ onto some open subset of $M$.
Even with all of this, I still have not emphasized strongly enough the "group theoretic" feature of the solution curves, i.e. the equation
$$phi_t(phi_s(x)) = phi_{t+s}(x)
$$
In the global case this equation is true for all $x in M$ and all $s,t in mathbb R$.
In the local case this equation is true if everything is defined, namely: if $x in U$ and $s in (-epsilon,+epsilon)$ hence $phi_s(x)$ is defined; and if $phi_s(x) in U$ and $t in (-epsilon,+epsilon)$ hence $phi_t(phi_s(x))$ is defined; and if $t+s in (-epsilon,+epsilon)$ hence $phi_{t+s}(x)$ is defined, then the equation is true.
Your "extra question" is answered by the Cartan Lie theorem.
answered Feb 1 at 13:59
Lee MosherLee Mosher
51.8k33889
$endgroup$
$begingroup$
Thanks for your time! it helped me a lot!
$endgroup$
– Hurjui Ionut
Feb 4 at 18:18
$begingroup$
Glad to hear it!
$endgroup$
– Lee Mosher
Feb 4 at 22:42
add a comment |
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$begingroup$
Loosely speaking, "global" and "local" in this context simply record whether the expression $phi_t(x)$ in question is defined for all $x in M$ and all $t in mathbb R$, or instead just for $x,t$ in some particular open set (I'm using an augmented notation $phi_t(x)$ where $x$ is put in as a function parameter to represent the initial condition).
For example, the hypothesis of the theorem refers to vector fields $X$ which generate "global 1-parameter groups $phi_t = exp tX$ of $M$", meaning that the solution curves of the differential equation $frac{dphi_t}{dt} = X_{phi_t}$ are defined for all initial conditions $x in M$ and for all $t in mathbb R$.
On the other hand, in general (as I'm sure you know) a vector field $X$ does not generate global 1-parameter groups. Instead, the actual conclusions of the existence/uniqueness theorem for solutions are often recorded using "local" terminology. To say that $X$ generates "local 1-parameter groups of local transformations of $M$" is simply to write down the conclusions of the existence/uniqueness theorem, or perhaps to write them down in a particular manner which emphasizes the "localness". Here's one such way to write the conclusions:
For each $x in M$ there exists an open neighborhood $U subset M$ of $x$ and an open interval $(-epsilon,+epsilon)$ with the following properties:
For each $y in U$ the solution curve $t mapsto phi_{t}(y)$ exists for all $t in (-epsilon,+epsilon)$, and it satisfies the condition that $frac{d}{dt}(phi_t(y)) = X_{phi_t(y)}$.
The map $U times (-epsilon,+epsilon) mapsto M$ defined by $(y,t) mapsto phi_t(y)$ is smooth.
For each fixed $t in (-epsilon,+epsilon)$ the map $U to M$ defined by $y mapsto phi_t(y)$ is a diffeomorphism from $U$ onto some open subset of $M$.
Even with all of this, I still have not emphasized strongly enough the "group theoretic" feature of the solution curves, i.e. the equation
$$phi_t(phi_s(x)) = phi_{t+s}(x)
$$
In the global case this equation is true for all $x in M$ and all $s,t in mathbb R$.
In the local case this equation is true if everything is defined, namely: if $x in U$ and $s in (-epsilon,+epsilon)$ hence $phi_s(x)$ is defined; and if $phi_s(x) in U$ and $t in (-epsilon,+epsilon)$ hence $phi_t(phi_s(x))$ is defined; and if $t+s in (-epsilon,+epsilon)$ hence $phi_{t+s}(x)$ is defined, then the equation is true.
Your "extra question" is answered by the Cartan Lie theorem.
answered Feb 1 at 13:59
Lee MosherLee Mosher
51.8k33889
$endgroup$
$begingroup$
Thanks for your time! it helped me a lot!
$endgroup$
– Hurjui Ionut
Feb 4 at 18:18
$begingroup$
Glad to hear it!
$endgroup$
– Lee Mosher
Feb 4 at 22:42
add a comment |
$begingroup$
Loosely speaking, "global" and "local" in this context simply record whether the expression $phi_t(x)$ in question is defined for all $x in M$ and all $t in mathbb R$, or instead just for $x,t$ in some particular open set (I'm using an augmented notation $phi_t(x)$ where $x$ is put in as a function parameter to represent the initial condition).
For example, the hypothesis of the theorem refers to vector fields $X$ which generate "global 1-parameter groups $phi_t = exp tX$ of $M$", meaning that the solution curves of the differential equation $frac{dphi_t}{dt} = X_{phi_t}$ are defined for all initial conditions $x in M$ and for all $t in mathbb R$.
On the other hand, in general (as I'm sure you know) a vector field $X$ does not generate global 1-parameter groups. Instead, the actual conclusions of the existence/uniqueness theorem for solutions are often recorded using "local" terminology. To say that $X$ generates "local 1-parameter groups of local transformations of $M$" is simply to write down the conclusions of the existence/uniqueness theorem, or perhaps to write them down in a particular manner which emphasizes the "localness". Here's one such way to write the conclusions:
For each $x in M$ there exists an open neighborhood $U subset M$ of $x$ and an open interval $(-epsilon,+epsilon)$ with the following properties:
For each $y in U$ the solution curve $t mapsto phi_{t}(y)$ exists for all $t in (-epsilon,+epsilon)$, and it satisfies the condition that $frac{d}{dt}(phi_t(y)) = X_{phi_t(y)}$.
The map $U times (-epsilon,+epsilon) mapsto M$ defined by $(y,t) mapsto phi_t(y)$ is smooth.
For each fixed $t in (-epsilon,+epsilon)$ the map $U to M$ defined by $y mapsto phi_t(y)$ is a diffeomorphism from $U$ onto some open subset of $M$.
Even with all of this, I still have not emphasized strongly enough the "group theoretic" feature of the solution curves, i.e. the equation
$$phi_t(phi_s(x)) = phi_{t+s}(x)
$$
In the global case this equation is true for all $x in M$ and all $s,t in mathbb R$.
In the local case this equation is true if everything is defined, namely: if $x in U$ and $s in (-epsilon,+epsilon)$ hence $phi_s(x)$ is defined; and if $phi_s(x) in U$ and $t in (-epsilon,+epsilon)$ hence $phi_t(phi_s(x))$ is defined; and if $t+s in (-epsilon,+epsilon)$ hence $phi_{t+s}(x)$ is defined, then the equation is true.
Your "extra question" is answered by the Cartan Lie theorem.
answered Feb 1 at 13:59
Lee MosherLee Mosher
51.8k33889
$endgroup$
$begingroup$
Thanks for your time! it helped me a lot!
$endgroup$
– Hurjui Ionut
Feb 4 at 18:18
$begingroup$
Glad to hear it!
$endgroup$
– Lee Mosher
Feb 4 at 22:42
add a comment |
$begingroup$
Loosely speaking, "global" and "local" in this context simply record whether the expression $phi_t(x)$ in question is defined for all $x in M$ and all $t in mathbb R$, or instead just for $x,t$ in some particular open set (I'm using an augmented notation $phi_t(x)$ where $x$ is put in as a function parameter to represent the initial condition).
For example, the hypothesis of the theorem refers to vector fields $X$ which generate "global 1-parameter groups $phi_t = exp tX$ of $M$", meaning that the solution curves of the differential equation $frac{dphi_t}{dt} = X_{phi_t}$ are defined for all initial conditions $x in M$ and for all $t in mathbb R$.
On the other hand, in general (as I'm sure you know) a vector field $X$ does not generate global 1-parameter groups. Instead, the actual conclusions of the existence/uniqueness theorem for solutions are often recorded using "local" terminology. To say that $X$ generates "local 1-parameter groups of local transformations of $M$" is simply to write down the conclusions of the existence/uniqueness theorem, or perhaps to write them down in a particular manner which emphasizes the "localness". Here's one such way to write the conclusions:
For each $x in M$ there exists an open neighborhood $U subset M$ of $x$ and an open interval $(-epsilon,+epsilon)$ with the following properties:
For each $y in U$ the solution curve $t mapsto phi_{t}(y)$ exists for all $t in (-epsilon,+epsilon)$, and it satisfies the condition that $frac{d}{dt}(phi_t(y)) = X_{phi_t(y)}$.
The map $U times (-epsilon,+epsilon) mapsto M$ defined by $(y,t) mapsto phi_t(y)$ is smooth.
For each fixed $t in (-epsilon,+epsilon)$ the map $U to M$ defined by $y mapsto phi_t(y)$ is a diffeomorphism from $U$ onto some open subset of $M$.
Even with all of this, I still have not emphasized strongly enough the "group theoretic" feature of the solution curves, i.e. the equation
$$phi_t(phi_s(x)) = phi_{t+s}(x)
$$
In the global case this equation is true for all $x in M$ and all $s,t in mathbb R$.
In the local case this equation is true if everything is defined, namely: if $x in U$ and $s in (-epsilon,+epsilon)$ hence $phi_s(x)$ is defined; and if $phi_s(x) in U$ and $t in (-epsilon,+epsilon)$ hence $phi_t(phi_s(x))$ is defined; and if $t+s in (-epsilon,+epsilon)$ hence $phi_{t+s}(x)$ is defined, then the equation is true.
Your "extra question" is answered by the Cartan Lie theorem.
answered Feb 1 at 13:59
Lee MosherLee Mosher
51.8k33889
$endgroup$
Loosely speaking, "global" and "local" in this context simply record whether the expression $phi_t(x)$ in question is defined for all $x in M$ and all $t in mathbb R$, or instead just for $x,t$ in some particular open set (I'm using an augmented notation $phi_t(x)$ where $x$ is put in as a function parameter to represent the initial condition).
For example, the hypothesis of the theorem refers to vector fields $X$ which generate "global 1-parameter groups $phi_t = exp tX$ of $M$", meaning that the solution curves of the differential equation $frac{dphi_t}{dt} = X_{phi_t}$ are defined for all initial conditions $x in M$ and for all $t in mathbb R$.
On the other hand, in general (as I'm sure you know) a vector field $X$ does not generate global 1-parameter groups. Instead, the actual conclusions of the existence/uniqueness theorem for solutions are often recorded using "local" terminology. To say that $X$ generates "local 1-parameter groups of local transformations of $M$" is simply to write down the conclusions of the existence/uniqueness theorem, or perhaps to write them down in a particular manner which emphasizes the "localness". Here's one such way to write the conclusions:
For each $x in M$ there exists an open neighborhood $U subset M$ of $x$ and an open interval $(-epsilon,+epsilon)$ with the following properties:
For each $y in U$ the solution curve $t mapsto phi_{t}(y)$ exists for all $t in (-epsilon,+epsilon)$, and it satisfies the condition that $frac{d}{dt}(phi_t(y)) = X_{phi_t(y)}$.
The map $U times (-epsilon,+epsilon) mapsto M$ defined by $(y,t) mapsto phi_t(y)$ is smooth.
For each fixed $t in (-epsilon,+epsilon)$ the map $U to M$ defined by $y mapsto phi_t(y)$ is a diffeomorphism from $U$ onto some open subset of $M$.
Even with all of this, I still have not emphasized strongly enough the "group theoretic" feature of the solution curves, i.e. the equation
$$phi_t(phi_s(x)) = phi_{t+s}(x)
$$
In the global case this equation is true for all $x in M$ and all $s,t in mathbb R$.
In the local case this equation is true if everything is defined, namely: if $x in U$ and $s in (-epsilon,+epsilon)$ hence $phi_s(x)$ is defined; and if $phi_s(x) in U$ and $t in (-epsilon,+epsilon)$ hence $phi_t(phi_s(x))$ is defined; and if $t+s in (-epsilon,+epsilon)$ hence $phi_{t+s}(x)$ is defined, then the equation is true.
Your "extra question" is answered by the Cartan Lie theorem.
answered Feb 1 at 13:59
Lee MosherLee Mosher
51.8k33889
edited Feb 1 at 14:29
answered Feb 1 at 13:59
Lee MosherLee Mosher
51.8k33889
answered Feb 1 at 13:59
Lee MosherLee Mosher
51.8k33889
answered Feb 1 at 13:59
Lee MosherLee Mosher
51.8k33889
51.8k33889
$begingroup$
Thanks for your time! it helped me a lot!
$endgroup$
– Hurjui Ionut
Feb 4 at 18:18
$begingroup$
Glad to hear it!
$endgroup$
– Lee Mosher
Feb 4 at 22:42
add a comment |
$begingroup$
Thanks for your time! it helped me a lot!
$endgroup$
– Hurjui Ionut
Feb 4 at 18:18
$begingroup$
Glad to hear it!
$endgroup$
– Lee Mosher
Feb 4 at 22:42
$begingroup$
Thanks for your time! it helped me a lot!
$endgroup$
– Hurjui Ionut
Feb 4 at 18:18
$begingroup$
Thanks for your time! it helped me a lot!
$endgroup$
– Hurjui Ionut
Feb 4 at 18:18
$begingroup$
Glad to hear it!
$endgroup$
– Lee Mosher
Feb 4 at 22:42
$begingroup$
Glad to hear it!
$endgroup$
– Lee Mosher
Feb 4 at 22:42
add a comment |
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