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Functions on a manifold
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I am learning about diff geometry and I have a question about ways of defining a function on a manifold. So if we think about manifolds in general we can define a chart on it. Then, through composition of a function and a chart we can analyze this function. My question is this. How can we define a function in the first place since it is on points P which need not be numbers. Dont we need to define the function on a chart in order for it to be defined on a manifold? And dont we then need to define a manifold through charts also? But in order for it to be defined properly we have to look for chart independent ptoperties which are read off the charts... I am confused.
differential-geometry manifolds
asked 2 days ago
Žarko Tomičić
1012
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Žarko Tomičić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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I am learning about diff geometry and I have a question about ways of defining a function on a manifold. So if we think about manifolds in general we can define a chart on it. Then, through composition of a function and a chart we can analyze this function. My question is this. How can we define a function in the first place since it is on points P which need not be numbers. Dont we need to define the function on a chart in order for it to be defined on a manifold? And dont we then need to define a manifold through charts also? But in order for it to be defined properly we have to look for chart independent ptoperties which are read off the charts... I am confused.
differential-geometry manifolds
asked 2 days ago
Žarko Tomičić
1012
New contributor
Žarko Tomičić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
What is a definition of function?
– Anubhav Mukherjee
2 days ago
@Anubhav Mukherjee Map from manifold to R.
– Žarko Tomičić
2 days ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am learning about diff geometry and I have a question about ways of defining a function on a manifold. So if we think about manifolds in general we can define a chart on it. Then, through composition of a function and a chart we can analyze this function. My question is this. How can we define a function in the first place since it is on points P which need not be numbers. Dont we need to define the function on a chart in order for it to be defined on a manifold? And dont we then need to define a manifold through charts also? But in order for it to be defined properly we have to look for chart independent ptoperties which are read off the charts... I am confused.
differential-geometry manifolds
asked 2 days ago
Žarko Tomičić
1012
New contributor
Žarko Tomičić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I am learning about diff geometry and I have a question about ways of defining a function on a manifold. So if we think about manifolds in general we can define a chart on it. Then, through composition of a function and a chart we can analyze this function. My question is this. How can we define a function in the first place since it is on points P which need not be numbers. Dont we need to define the function on a chart in order for it to be defined on a manifold? And dont we then need to define a manifold through charts also? But in order for it to be defined properly we have to look for chart independent ptoperties which are read off the charts... I am confused.
differential-geometry manifolds
differential-geometry manifolds
asked 2 days ago
Žarko Tomičić
1012
New contributor
Žarko Tomičić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
Žarko Tomičić
1012
New contributor
Žarko Tomičić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
Žarko Tomičić
1012
New contributor
Žarko Tomičić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
Žarko Tomičić
1012
asked 2 days ago
Žarko Tomičić
1012
1012
New contributor
Žarko Tomičić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Žarko Tomičić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Žarko Tomičić is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
What is a definition of function?
– Anubhav Mukherjee
2 days ago
@Anubhav Mukherjee Map from manifold to R.
– Žarko Tomičić
2 days ago
add a comment |
What is a definition of function?
– Anubhav Mukherjee
2 days ago
@Anubhav Mukherjee Map from manifold to R.
– Žarko Tomičić
2 days ago
What is a definition of function?
– Anubhav Mukherjee
2 days ago
What is a definition of function?
– Anubhav Mukherjee
2 days ago
@Anubhav Mukherjee Map from manifold to R.
– Žarko Tomičić
2 days ago
@Anubhav Mukherjee Map from manifold to R.
– Žarko Tomičić
2 days ago
add a comment |
2 Answers
2
active
oldest
votes
up vote
2
down vote
I like your question, since I believe it is connected to one of the core topics of differential geometry.
Being not an expert, I can only give you my humble opinion on it, which is better expressed by a few questions, rather than long answers...
Abstract vs. concrete functions...
For sets $X, Y$, a function $f: X to Y$ assigns a value $f(x)$ for each $x in X$.
But if $X$ is large, we may want to use efficient ways to express $f(x)$ in terms of $x$.
To define a function on a manifold, we just need to assignment a value $f(p)$ for all $p in mathcal M$. (Which maybe also satisfied other conditions, e.g. smoothness.) If such an assignment is given, it will be automatically coordinate-free, since we did never toughed the charts to define it. That's the reason why we like this approach for abstract purposes.
But since our human brain has only finite memory,
this method is a bit unpractical if we want to do calculations etc.
How can we practically define functions on manifolds
Gobal chart: If we have a global chart $h: mathcal M to mathbb R^n$, then we can define a function $f = tilde f circ h$, with $tilde f:mathbb R^n to mathbb R$ being a smooth function.
Similary, we might use embeddings $e: mathcal M to mathbb R^N$ to define a function for example on submanifolds of $mathbb R^N$.
This is for example commonly used to define maps on submanifolds ($mathrm{GL}(n)$, $mathrm{SO}(n)$, ...) of the manifold of matrices $mathrm{Mat}(ntimes n)$.
Finitely many charts: For two charts $h_1: mathcal U to mathbb R^n$ and $h_2: mathcal V to mathbb R^n$, we may define $f_1 : mathcal U to mathbb R$ and $f_2 : mathcal V to mathbb R$ as above, but then $f_1 = f_2$ must be assured on the intersection of the chart domains, i.e. for all $p in mathcal U cap mathcal V$.
Implicit definitions:
It is sometimes still possible to calculate on manifolds, without having an explicit formula for a function.
Solutions of (partial) differential equations:
Assume, we can define a differential equation on a manifold.
For example the Laplace equation $Delta u = 0$ (with some boundary conditions, for example constant values on 'some' boundaries). Then we can define a map $u(p)$, by claiming that it is the unique solution of the partial differential equation. In particular, it is sufficient to provide a finite amount of data, to define the function! If the manifold has two boundaries, we just need to pick two numbers and $u$ is defined (if the PDE is well-defined, of course)!
With this function $u$, we can continue to work, since we have (implicit) information about the function $u$ as well, which is sufficient for many tasks.
But the examples don't stop here: We have the geodesic equation (and ODE), or Hamiltons equations etc...
Of course there is a hidden trap: To define a differential equation, say $dot x(t) = F(x(t))$ on a manifold, we need a vector field $F$, which is
again a function $F: mathcal M to T mathcal M$.
But in important special cases, we know a little bit about $T mathcal M$, which allows us to define, maybe implicitly, interesting functions of that type.
(For example if $mathcal M$ is a Lie group, then we can define one value $F(1) in T_1 mathcal M$. Now Lie groups are super cool and for each point $p$ there is a meaningful map $T lambda_p : T_1 mathcal M to T_p mathcal M$. This allows us to define $F(p) = D lambda_p F(1)$. Expressed in coordinates, this map may not look like a constant map, and the solutions $x(t)$ are also somehow interesting.)
At the end...
It is good to ask these equations! And be patient with the theory. Differential geometry need some time to feel natural, but after a while you will see that numbers, charts and coordinates are sometimes false friends. Spaces are often quite nonlinear!
EDIT:
Concrete Example:
We use a function $f: mathbb R^3 to mathbb R: (x,y,z) mapsto x^2+y^4+z^2$.
Then we define $mathcal M := f^{-1}({1})$. This is indeed a manifold by the constant rank theorem, since $mathrm{D} f(p) = 1$, for all points $p in mathcal M$.
But we also have an embedding $mathcal M to mathbb R^3: p=(x,y,z) mapsto (x,y,z)$.
Therefore I can choose a function on $mathbb R^3$, say
$$
tilde f(x,y,z) = sin(x) cdot exp(y)^{z^2}
$$
and define
$$
f(p) = tilde f circ e = tilde f(x,y,z).
$$
As you see, we did not directly use the charts of $mathcal M$, but only implicit information about $mathcal M$.
answered 2 days ago
Steffen Plunder
488211
Thank you for your answer man...so, would you say that abstract manifold is actually defined through its charts?
– Žarko Tomičić
2 days ago
Because, you see, I have a problem. I used to think its all clear, you know, you define some coordinet axes and than put some numbers on space points, in 3d physical space eg. But then it ocured to me, where and how should I put the axes and what are these axes exactly?
– Žarko Tomičić
2 days ago
Even more so, if you try to chart a space time manifold which is actually our physical space....torsion free, 4d manifold lalaalala...
– Žarko Tomičić
2 days ago
I added a concrete example, which does not use charts directly. About the charts: (I agree with the replies from Federico.) Yes, charts are part of the differentiable structure, which is needed to define a manifold. But, we do not need to know them explicitly and even if, the idea is to think about points of a manifold to be not just points, which we can described in a linear manner by a few numbers. About physical spaces: Is there one or are there many (equally important) charts for physical spaces? Some calculus laws with x-y-z coordinates are simply not true for general charts.
– Steffen Plunder
yesterday
add a comment |
up vote
0
down vote
How can we define a function in the first place since it is on points P which need not be numbers.
Functions need not be defined only on "numbers", whatever that means.
Functions can be defined on arbitrary sets, in particular manifolds. For instance, given a manifold $M$, you can consider $f:Mtomathbb R$ defined as $f(p)=0$ for every $pin M$. Also notice that charts are actually functions $Msupset Utomathbb R^d$, so I don't understand why you are ok with them but not with other functions.
answered 2 days ago
Federico
2,05658
yes I agree that you can say it is zero everywhere, but that is not the point. How can you define in an exact way something more complicated?
– Žarko Tomičić
2 days ago
What do you mean? $M^S={f:M to S}$ is a well defined set for every $M$ and $S$.
– Federico
2 days ago
Regarding charts, yes, I am aware of them being maps from a manifold M to Rd, but not to R. Nevertheless, it still confuses me. How exactly do you chart a manifold?
– Žarko Tomičić
2 days ago
There can be plenty of ways to describe any specific function.
– Federico
2 days ago
For instance, $S^2subsetmathbb R^3$ can be endowed with a manifold structure. But since it is also already embedded inside $mathbb R^3$, you can also just take any function defined on the ambient space and restrict it.
– Federico
2 days ago
|
show 12 more comments
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
I like your question, since I believe it is connected to one of the core topics of differential geometry.
Being not an expert, I can only give you my humble opinion on it, which is better expressed by a few questions, rather than long answers...
Abstract vs. concrete functions...
For sets $X, Y$, a function $f: X to Y$ assigns a value $f(x)$ for each $x in X$.
But if $X$ is large, we may want to use efficient ways to express $f(x)$ in terms of $x$.
To define a function on a manifold, we just need to assignment a value $f(p)$ for all $p in mathcal M$. (Which maybe also satisfied other conditions, e.g. smoothness.) If such an assignment is given, it will be automatically coordinate-free, since we did never toughed the charts to define it. That's the reason why we like this approach for abstract purposes.
But since our human brain has only finite memory,
this method is a bit unpractical if we want to do calculations etc.
How can we practically define functions on manifolds
Gobal chart: If we have a global chart $h: mathcal M to mathbb R^n$, then we can define a function $f = tilde f circ h$, with $tilde f:mathbb R^n to mathbb R$ being a smooth function.
Similary, we might use embeddings $e: mathcal M to mathbb R^N$ to define a function for example on submanifolds of $mathbb R^N$.
This is for example commonly used to define maps on submanifolds ($mathrm{GL}(n)$, $mathrm{SO}(n)$, ...) of the manifold of matrices $mathrm{Mat}(ntimes n)$.
Finitely many charts: For two charts $h_1: mathcal U to mathbb R^n$ and $h_2: mathcal V to mathbb R^n$, we may define $f_1 : mathcal U to mathbb R$ and $f_2 : mathcal V to mathbb R$ as above, but then $f_1 = f_2$ must be assured on the intersection of the chart domains, i.e. for all $p in mathcal U cap mathcal V$.
Implicit definitions:
It is sometimes still possible to calculate on manifolds, without having an explicit formula for a function.
Solutions of (partial) differential equations:
Assume, we can define a differential equation on a manifold.
For example the Laplace equation $Delta u = 0$ (with some boundary conditions, for example constant values on 'some' boundaries). Then we can define a map $u(p)$, by claiming that it is the unique solution of the partial differential equation. In particular, it is sufficient to provide a finite amount of data, to define the function! If the manifold has two boundaries, we just need to pick two numbers and $u$ is defined (if the PDE is well-defined, of course)!
With this function $u$, we can continue to work, since we have (implicit) information about the function $u$ as well, which is sufficient for many tasks.
But the examples don't stop here: We have the geodesic equation (and ODE), or Hamiltons equations etc...
Of course there is a hidden trap: To define a differential equation, say $dot x(t) = F(x(t))$ on a manifold, we need a vector field $F$, which is
again a function $F: mathcal M to T mathcal M$.
But in important special cases, we know a little bit about $T mathcal M$, which allows us to define, maybe implicitly, interesting functions of that type.
(For example if $mathcal M$ is a Lie group, then we can define one value $F(1) in T_1 mathcal M$. Now Lie groups are super cool and for each point $p$ there is a meaningful map $T lambda_p : T_1 mathcal M to T_p mathcal M$. This allows us to define $F(p) = D lambda_p F(1)$. Expressed in coordinates, this map may not look like a constant map, and the solutions $x(t)$ are also somehow interesting.)
At the end...
It is good to ask these equations! And be patient with the theory. Differential geometry need some time to feel natural, but after a while you will see that numbers, charts and coordinates are sometimes false friends. Spaces are often quite nonlinear!
EDIT:
Concrete Example:
We use a function $f: mathbb R^3 to mathbb R: (x,y,z) mapsto x^2+y^4+z^2$.
Then we define $mathcal M := f^{-1}({1})$. This is indeed a manifold by the constant rank theorem, since $mathrm{D} f(p) = 1$, for all points $p in mathcal M$.
But we also have an embedding $mathcal M to mathbb R^3: p=(x,y,z) mapsto (x,y,z)$.
Therefore I can choose a function on $mathbb R^3$, say
$$
tilde f(x,y,z) = sin(x) cdot exp(y)^{z^2}
$$
and define
$$
f(p) = tilde f circ e = tilde f(x,y,z).
$$
As you see, we did not directly use the charts of $mathcal M$, but only implicit information about $mathcal M$.
answered 2 days ago
Steffen Plunder
488211
Thank you for your answer man...so, would you say that abstract manifold is actually defined through its charts?
– Žarko Tomičić
2 days ago
Because, you see, I have a problem. I used to think its all clear, you know, you define some coordinet axes and than put some numbers on space points, in 3d physical space eg. But then it ocured to me, where and how should I put the axes and what are these axes exactly?
– Žarko Tomičić
2 days ago
Even more so, if you try to chart a space time manifold which is actually our physical space....torsion free, 4d manifold lalaalala...
– Žarko Tomičić
2 days ago
I added a concrete example, which does not use charts directly. About the charts: (I agree with the replies from Federico.) Yes, charts are part of the differentiable structure, which is needed to define a manifold. But, we do not need to know them explicitly and even if, the idea is to think about points of a manifold to be not just points, which we can described in a linear manner by a few numbers. About physical spaces: Is there one or are there many (equally important) charts for physical spaces? Some calculus laws with x-y-z coordinates are simply not true for general charts.
– Steffen Plunder
yesterday
add a comment |
up vote
2
down vote
I like your question, since I believe it is connected to one of the core topics of differential geometry.
Being not an expert, I can only give you my humble opinion on it, which is better expressed by a few questions, rather than long answers...
Abstract vs. concrete functions...
For sets $X, Y$, a function $f: X to Y$ assigns a value $f(x)$ for each $x in X$.
But if $X$ is large, we may want to use efficient ways to express $f(x)$ in terms of $x$.
To define a function on a manifold, we just need to assignment a value $f(p)$ for all $p in mathcal M$. (Which maybe also satisfied other conditions, e.g. smoothness.) If such an assignment is given, it will be automatically coordinate-free, since we did never toughed the charts to define it. That's the reason why we like this approach for abstract purposes.
But since our human brain has only finite memory,
this method is a bit unpractical if we want to do calculations etc.
How can we practically define functions on manifolds
Gobal chart: If we have a global chart $h: mathcal M to mathbb R^n$, then we can define a function $f = tilde f circ h$, with $tilde f:mathbb R^n to mathbb R$ being a smooth function.
Similary, we might use embeddings $e: mathcal M to mathbb R^N$ to define a function for example on submanifolds of $mathbb R^N$.
This is for example commonly used to define maps on submanifolds ($mathrm{GL}(n)$, $mathrm{SO}(n)$, ...) of the manifold of matrices $mathrm{Mat}(ntimes n)$.
Finitely many charts: For two charts $h_1: mathcal U to mathbb R^n$ and $h_2: mathcal V to mathbb R^n$, we may define $f_1 : mathcal U to mathbb R$ and $f_2 : mathcal V to mathbb R$ as above, but then $f_1 = f_2$ must be assured on the intersection of the chart domains, i.e. for all $p in mathcal U cap mathcal V$.
Implicit definitions:
It is sometimes still possible to calculate on manifolds, without having an explicit formula for a function.
Solutions of (partial) differential equations:
Assume, we can define a differential equation on a manifold.
For example the Laplace equation $Delta u = 0$ (with some boundary conditions, for example constant values on 'some' boundaries). Then we can define a map $u(p)$, by claiming that it is the unique solution of the partial differential equation. In particular, it is sufficient to provide a finite amount of data, to define the function! If the manifold has two boundaries, we just need to pick two numbers and $u$ is defined (if the PDE is well-defined, of course)!
With this function $u$, we can continue to work, since we have (implicit) information about the function $u$ as well, which is sufficient for many tasks.
But the examples don't stop here: We have the geodesic equation (and ODE), or Hamiltons equations etc...
Of course there is a hidden trap: To define a differential equation, say $dot x(t) = F(x(t))$ on a manifold, we need a vector field $F$, which is
again a function $F: mathcal M to T mathcal M$.
But in important special cases, we know a little bit about $T mathcal M$, which allows us to define, maybe implicitly, interesting functions of that type.
(For example if $mathcal M$ is a Lie group, then we can define one value $F(1) in T_1 mathcal M$. Now Lie groups are super cool and for each point $p$ there is a meaningful map $T lambda_p : T_1 mathcal M to T_p mathcal M$. This allows us to define $F(p) = D lambda_p F(1)$. Expressed in coordinates, this map may not look like a constant map, and the solutions $x(t)$ are also somehow interesting.)
At the end...
It is good to ask these equations! And be patient with the theory. Differential geometry need some time to feel natural, but after a while you will see that numbers, charts and coordinates are sometimes false friends. Spaces are often quite nonlinear!
EDIT:
Concrete Example:
We use a function $f: mathbb R^3 to mathbb R: (x,y,z) mapsto x^2+y^4+z^2$.
Then we define $mathcal M := f^{-1}({1})$. This is indeed a manifold by the constant rank theorem, since $mathrm{D} f(p) = 1$, for all points $p in mathcal M$.
But we also have an embedding $mathcal M to mathbb R^3: p=(x,y,z) mapsto (x,y,z)$.
Therefore I can choose a function on $mathbb R^3$, say
$$
tilde f(x,y,z) = sin(x) cdot exp(y)^{z^2}
$$
and define
$$
f(p) = tilde f circ e = tilde f(x,y,z).
$$
As you see, we did not directly use the charts of $mathcal M$, but only implicit information about $mathcal M$.
answered 2 days ago
Steffen Plunder
488211
Thank you for your answer man...so, would you say that abstract manifold is actually defined through its charts?
– Žarko Tomičić
2 days ago
Because, you see, I have a problem. I used to think its all clear, you know, you define some coordinet axes and than put some numbers on space points, in 3d physical space eg. But then it ocured to me, where and how should I put the axes and what are these axes exactly?
– Žarko Tomičić
2 days ago
Even more so, if you try to chart a space time manifold which is actually our physical space....torsion free, 4d manifold lalaalala...
– Žarko Tomičić
2 days ago
I added a concrete example, which does not use charts directly. About the charts: (I agree with the replies from Federico.) Yes, charts are part of the differentiable structure, which is needed to define a manifold. But, we do not need to know them explicitly and even if, the idea is to think about points of a manifold to be not just points, which we can described in a linear manner by a few numbers. About physical spaces: Is there one or are there many (equally important) charts for physical spaces? Some calculus laws with x-y-z coordinates are simply not true for general charts.
– Steffen Plunder
yesterday
add a comment |
up vote
2
down vote
up vote
2
down vote
I like your question, since I believe it is connected to one of the core topics of differential geometry.
Being not an expert, I can only give you my humble opinion on it, which is better expressed by a few questions, rather than long answers...
Abstract vs. concrete functions...
For sets $X, Y$, a function $f: X to Y$ assigns a value $f(x)$ for each $x in X$.
But if $X$ is large, we may want to use efficient ways to express $f(x)$ in terms of $x$.
To define a function on a manifold, we just need to assignment a value $f(p)$ for all $p in mathcal M$. (Which maybe also satisfied other conditions, e.g. smoothness.) If such an assignment is given, it will be automatically coordinate-free, since we did never toughed the charts to define it. That's the reason why we like this approach for abstract purposes.
But since our human brain has only finite memory,
this method is a bit unpractical if we want to do calculations etc.
How can we practically define functions on manifolds
Gobal chart: If we have a global chart $h: mathcal M to mathbb R^n$, then we can define a function $f = tilde f circ h$, with $tilde f:mathbb R^n to mathbb R$ being a smooth function.
Similary, we might use embeddings $e: mathcal M to mathbb R^N$ to define a function for example on submanifolds of $mathbb R^N$.
This is for example commonly used to define maps on submanifolds ($mathrm{GL}(n)$, $mathrm{SO}(n)$, ...) of the manifold of matrices $mathrm{Mat}(ntimes n)$.
Finitely many charts: For two charts $h_1: mathcal U to mathbb R^n$ and $h_2: mathcal V to mathbb R^n$, we may define $f_1 : mathcal U to mathbb R$ and $f_2 : mathcal V to mathbb R$ as above, but then $f_1 = f_2$ must be assured on the intersection of the chart domains, i.e. for all $p in mathcal U cap mathcal V$.
Implicit definitions:
It is sometimes still possible to calculate on manifolds, without having an explicit formula for a function.
Solutions of (partial) differential equations:
Assume, we can define a differential equation on a manifold.
For example the Laplace equation $Delta u = 0$ (with some boundary conditions, for example constant values on 'some' boundaries). Then we can define a map $u(p)$, by claiming that it is the unique solution of the partial differential equation. In particular, it is sufficient to provide a finite amount of data, to define the function! If the manifold has two boundaries, we just need to pick two numbers and $u$ is defined (if the PDE is well-defined, of course)!
With this function $u$, we can continue to work, since we have (implicit) information about the function $u$ as well, which is sufficient for many tasks.
But the examples don't stop here: We have the geodesic equation (and ODE), or Hamiltons equations etc...
Of course there is a hidden trap: To define a differential equation, say $dot x(t) = F(x(t))$ on a manifold, we need a vector field $F$, which is
again a function $F: mathcal M to T mathcal M$.
But in important special cases, we know a little bit about $T mathcal M$, which allows us to define, maybe implicitly, interesting functions of that type.
(For example if $mathcal M$ is a Lie group, then we can define one value $F(1) in T_1 mathcal M$. Now Lie groups are super cool and for each point $p$ there is a meaningful map $T lambda_p : T_1 mathcal M to T_p mathcal M$. This allows us to define $F(p) = D lambda_p F(1)$. Expressed in coordinates, this map may not look like a constant map, and the solutions $x(t)$ are also somehow interesting.)
At the end...
It is good to ask these equations! And be patient with the theory. Differential geometry need some time to feel natural, but after a while you will see that numbers, charts and coordinates are sometimes false friends. Spaces are often quite nonlinear!
EDIT:
Concrete Example:
We use a function $f: mathbb R^3 to mathbb R: (x,y,z) mapsto x^2+y^4+z^2$.
Then we define $mathcal M := f^{-1}({1})$. This is indeed a manifold by the constant rank theorem, since $mathrm{D} f(p) = 1$, for all points $p in mathcal M$.
But we also have an embedding $mathcal M to mathbb R^3: p=(x,y,z) mapsto (x,y,z)$.
Therefore I can choose a function on $mathbb R^3$, say
$$
tilde f(x,y,z) = sin(x) cdot exp(y)^{z^2}
$$
and define
$$
f(p) = tilde f circ e = tilde f(x,y,z).
$$
As you see, we did not directly use the charts of $mathcal M$, but only implicit information about $mathcal M$.
answered 2 days ago
Steffen Plunder
488211
I like your question, since I believe it is connected to one of the core topics of differential geometry.
Being not an expert, I can only give you my humble opinion on it, which is better expressed by a few questions, rather than long answers...
Abstract vs. concrete functions...
For sets $X, Y$, a function $f: X to Y$ assigns a value $f(x)$ for each $x in X$.
But if $X$ is large, we may want to use efficient ways to express $f(x)$ in terms of $x$.
To define a function on a manifold, we just need to assignment a value $f(p)$ for all $p in mathcal M$. (Which maybe also satisfied other conditions, e.g. smoothness.) If such an assignment is given, it will be automatically coordinate-free, since we did never toughed the charts to define it. That's the reason why we like this approach for abstract purposes.
But since our human brain has only finite memory,
this method is a bit unpractical if we want to do calculations etc.
How can we practically define functions on manifolds
Gobal chart: If we have a global chart $h: mathcal M to mathbb R^n$, then we can define a function $f = tilde f circ h$, with $tilde f:mathbb R^n to mathbb R$ being a smooth function.
Similary, we might use embeddings $e: mathcal M to mathbb R^N$ to define a function for example on submanifolds of $mathbb R^N$.
This is for example commonly used to define maps on submanifolds ($mathrm{GL}(n)$, $mathrm{SO}(n)$, ...) of the manifold of matrices $mathrm{Mat}(ntimes n)$.
Finitely many charts: For two charts $h_1: mathcal U to mathbb R^n$ and $h_2: mathcal V to mathbb R^n$, we may define $f_1 : mathcal U to mathbb R$ and $f_2 : mathcal V to mathbb R$ as above, but then $f_1 = f_2$ must be assured on the intersection of the chart domains, i.e. for all $p in mathcal U cap mathcal V$.
Implicit definitions:
It is sometimes still possible to calculate on manifolds, without having an explicit formula for a function.
Solutions of (partial) differential equations:
Assume, we can define a differential equation on a manifold.
For example the Laplace equation $Delta u = 0$ (with some boundary conditions, for example constant values on 'some' boundaries). Then we can define a map $u(p)$, by claiming that it is the unique solution of the partial differential equation. In particular, it is sufficient to provide a finite amount of data, to define the function! If the manifold has two boundaries, we just need to pick two numbers and $u$ is defined (if the PDE is well-defined, of course)!
With this function $u$, we can continue to work, since we have (implicit) information about the function $u$ as well, which is sufficient for many tasks.
But the examples don't stop here: We have the geodesic equation (and ODE), or Hamiltons equations etc...
Of course there is a hidden trap: To define a differential equation, say $dot x(t) = F(x(t))$ on a manifold, we need a vector field $F$, which is
again a function $F: mathcal M to T mathcal M$.
But in important special cases, we know a little bit about $T mathcal M$, which allows us to define, maybe implicitly, interesting functions of that type.
(For example if $mathcal M$ is a Lie group, then we can define one value $F(1) in T_1 mathcal M$. Now Lie groups are super cool and for each point $p$ there is a meaningful map $T lambda_p : T_1 mathcal M to T_p mathcal M$. This allows us to define $F(p) = D lambda_p F(1)$. Expressed in coordinates, this map may not look like a constant map, and the solutions $x(t)$ are also somehow interesting.)
At the end...
It is good to ask these equations! And be patient with the theory. Differential geometry need some time to feel natural, but after a while you will see that numbers, charts and coordinates are sometimes false friends. Spaces are often quite nonlinear!
EDIT:
Concrete Example:
We use a function $f: mathbb R^3 to mathbb R: (x,y,z) mapsto x^2+y^4+z^2$.
Then we define $mathcal M := f^{-1}({1})$. This is indeed a manifold by the constant rank theorem, since $mathrm{D} f(p) = 1$, for all points $p in mathcal M$.
But we also have an embedding $mathcal M to mathbb R^3: p=(x,y,z) mapsto (x,y,z)$.
Therefore I can choose a function on $mathbb R^3$, say
$$
tilde f(x,y,z) = sin(x) cdot exp(y)^{z^2}
$$
and define
$$
f(p) = tilde f circ e = tilde f(x,y,z).
$$
As you see, we did not directly use the charts of $mathcal M$, but only implicit information about $mathcal M$.
answered 2 days ago
Steffen Plunder
488211
edited 2 days ago
answered 2 days ago
Steffen Plunder
488211
answered 2 days ago
Steffen Plunder
488211
answered 2 days ago
Steffen Plunder
488211
488211
Thank you for your answer man...so, would you say that abstract manifold is actually defined through its charts?
– Žarko Tomičić
2 days ago
Because, you see, I have a problem. I used to think its all clear, you know, you define some coordinet axes and than put some numbers on space points, in 3d physical space eg. But then it ocured to me, where and how should I put the axes and what are these axes exactly?
– Žarko Tomičić
2 days ago
Even more so, if you try to chart a space time manifold which is actually our physical space....torsion free, 4d manifold lalaalala...
– Žarko Tomičić
2 days ago
I added a concrete example, which does not use charts directly. About the charts: (I agree with the replies from Federico.) Yes, charts are part of the differentiable structure, which is needed to define a manifold. But, we do not need to know them explicitly and even if, the idea is to think about points of a manifold to be not just points, which we can described in a linear manner by a few numbers. About physical spaces: Is there one or are there many (equally important) charts for physical spaces? Some calculus laws with x-y-z coordinates are simply not true for general charts.
– Steffen Plunder
yesterday
add a comment |
Thank you for your answer man...so, would you say that abstract manifold is actually defined through its charts?
– Žarko Tomičić
2 days ago
Because, you see, I have a problem. I used to think its all clear, you know, you define some coordinet axes and than put some numbers on space points, in 3d physical space eg. But then it ocured to me, where and how should I put the axes and what are these axes exactly?
– Žarko Tomičić
2 days ago
Even more so, if you try to chart a space time manifold which is actually our physical space....torsion free, 4d manifold lalaalala...
– Žarko Tomičić
2 days ago
I added a concrete example, which does not use charts directly. About the charts: (I agree with the replies from Federico.) Yes, charts are part of the differentiable structure, which is needed to define a manifold. But, we do not need to know them explicitly and even if, the idea is to think about points of a manifold to be not just points, which we can described in a linear manner by a few numbers. About physical spaces: Is there one or are there many (equally important) charts for physical spaces? Some calculus laws with x-y-z coordinates are simply not true for general charts.
– Steffen Plunder
yesterday
Thank you for your answer man...so, would you say that abstract manifold is actually defined through its charts?
– Žarko Tomičić
2 days ago
Thank you for your answer man...so, would you say that abstract manifold is actually defined through its charts?
– Žarko Tomičić
2 days ago
Because, you see, I have a problem. I used to think its all clear, you know, you define some coordinet axes and than put some numbers on space points, in 3d physical space eg. But then it ocured to me, where and how should I put the axes and what are these axes exactly?
– Žarko Tomičić
2 days ago
Because, you see, I have a problem. I used to think its all clear, you know, you define some coordinet axes and than put some numbers on space points, in 3d physical space eg. But then it ocured to me, where and how should I put the axes and what are these axes exactly?
– Žarko Tomičić
2 days ago
Even more so, if you try to chart a space time manifold which is actually our physical space....torsion free, 4d manifold lalaalala...
– Žarko Tomičić
2 days ago
Even more so, if you try to chart a space time manifold which is actually our physical space....torsion free, 4d manifold lalaalala...
– Žarko Tomičić
2 days ago
I added a concrete example, which does not use charts directly. About the charts: (I agree with the replies from Federico.) Yes, charts are part of the differentiable structure, which is needed to define a manifold. But, we do not need to know them explicitly and even if, the idea is to think about points of a manifold to be not just points, which we can described in a linear manner by a few numbers. About physical spaces: Is there one or are there many (equally important) charts for physical spaces? Some calculus laws with x-y-z coordinates are simply not true for general charts.
– Steffen Plunder
yesterday
I added a concrete example, which does not use charts directly. About the charts: (I agree with the replies from Federico.) Yes, charts are part of the differentiable structure, which is needed to define a manifold. But, we do not need to know them explicitly and even if, the idea is to think about points of a manifold to be not just points, which we can described in a linear manner by a few numbers. About physical spaces: Is there one or are there many (equally important) charts for physical spaces? Some calculus laws with x-y-z coordinates are simply not true for general charts.
– Steffen Plunder
yesterday
add a comment |
up vote
0
down vote
How can we define a function in the first place since it is on points P which need not be numbers.
Functions need not be defined only on "numbers", whatever that means.
Functions can be defined on arbitrary sets, in particular manifolds. For instance, given a manifold $M$, you can consider $f:Mtomathbb R$ defined as $f(p)=0$ for every $pin M$. Also notice that charts are actually functions $Msupset Utomathbb R^d$, so I don't understand why you are ok with them but not with other functions.
answered 2 days ago
Federico
2,05658
yes I agree that you can say it is zero everywhere, but that is not the point. How can you define in an exact way something more complicated?
– Žarko Tomičić
2 days ago
What do you mean? $M^S={f:M to S}$ is a well defined set for every $M$ and $S$.
– Federico
2 days ago
Regarding charts, yes, I am aware of them being maps from a manifold M to Rd, but not to R. Nevertheless, it still confuses me. How exactly do you chart a manifold?
– Žarko Tomičić
2 days ago
There can be plenty of ways to describe any specific function.
– Federico
2 days ago
For instance, $S^2subsetmathbb R^3$ can be endowed with a manifold structure. But since it is also already embedded inside $mathbb R^3$, you can also just take any function defined on the ambient space and restrict it.
– Federico
2 days ago
|
show 12 more comments
up vote
0
down vote
How can we define a function in the first place since it is on points P which need not be numbers.
Functions need not be defined only on "numbers", whatever that means.
Functions can be defined on arbitrary sets, in particular manifolds. For instance, given a manifold $M$, you can consider $f:Mtomathbb R$ defined as $f(p)=0$ for every $pin M$. Also notice that charts are actually functions $Msupset Utomathbb R^d$, so I don't understand why you are ok with them but not with other functions.
answered 2 days ago
Federico
2,05658
yes I agree that you can say it is zero everywhere, but that is not the point. How can you define in an exact way something more complicated?
– Žarko Tomičić
2 days ago
What do you mean? $M^S={f:M to S}$ is a well defined set for every $M$ and $S$.
– Federico
2 days ago
Regarding charts, yes, I am aware of them being maps from a manifold M to Rd, but not to R. Nevertheless, it still confuses me. How exactly do you chart a manifold?
– Žarko Tomičić
2 days ago
There can be plenty of ways to describe any specific function.
– Federico
2 days ago
For instance, $S^2subsetmathbb R^3$ can be endowed with a manifold structure. But since it is also already embedded inside $mathbb R^3$, you can also just take any function defined on the ambient space and restrict it.
– Federico
2 days ago
|
show 12 more comments
up vote
0
down vote
up vote
0
down vote
How can we define a function in the first place since it is on points P which need not be numbers.
Functions need not be defined only on "numbers", whatever that means.
Functions can be defined on arbitrary sets, in particular manifolds. For instance, given a manifold $M$, you can consider $f:Mtomathbb R$ defined as $f(p)=0$ for every $pin M$. Also notice that charts are actually functions $Msupset Utomathbb R^d$, so I don't understand why you are ok with them but not with other functions.
answered 2 days ago
Federico
2,05658
How can we define a function in the first place since it is on points P which need not be numbers.
Functions need not be defined only on "numbers", whatever that means.
Functions can be defined on arbitrary sets, in particular manifolds. For instance, given a manifold $M$, you can consider $f:Mtomathbb R$ defined as $f(p)=0$ for every $pin M$. Also notice that charts are actually functions $Msupset Utomathbb R^d$, so I don't understand why you are ok with them but not with other functions.
answered 2 days ago
Federico
2,05658
answered 2 days ago
Federico
2,05658
answered 2 days ago
Federico
2,05658
answered 2 days ago
Federico
2,05658
2,05658
yes I agree that you can say it is zero everywhere, but that is not the point. How can you define in an exact way something more complicated?
– Žarko Tomičić
2 days ago
What do you mean? $M^S={f:M to S}$ is a well defined set for every $M$ and $S$.
– Federico
2 days ago
Regarding charts, yes, I am aware of them being maps from a manifold M to Rd, but not to R. Nevertheless, it still confuses me. How exactly do you chart a manifold?
– Žarko Tomičić
2 days ago
There can be plenty of ways to describe any specific function.
– Federico
2 days ago
For instance, $S^2subsetmathbb R^3$ can be endowed with a manifold structure. But since it is also already embedded inside $mathbb R^3$, you can also just take any function defined on the ambient space and restrict it.
– Federico
2 days ago
|
show 12 more comments
yes I agree that you can say it is zero everywhere, but that is not the point. How can you define in an exact way something more complicated?
– Žarko Tomičić
2 days ago
What do you mean? $M^S={f:M to S}$ is a well defined set for every $M$ and $S$.
– Federico
2 days ago
Regarding charts, yes, I am aware of them being maps from a manifold M to Rd, but not to R. Nevertheless, it still confuses me. How exactly do you chart a manifold?
– Žarko Tomičić
2 days ago
There can be plenty of ways to describe any specific function.
– Federico
2 days ago
For instance, $S^2subsetmathbb R^3$ can be endowed with a manifold structure. But since it is also already embedded inside $mathbb R^3$, you can also just take any function defined on the ambient space and restrict it.
– Federico
2 days ago
yes I agree that you can say it is zero everywhere, but that is not the point. How can you define in an exact way something more complicated?
– Žarko Tomičić
2 days ago
yes I agree that you can say it is zero everywhere, but that is not the point. How can you define in an exact way something more complicated?
– Žarko Tomičić
2 days ago
What do you mean? $M^S={f:M to S}$ is a well defined set for every $M$ and $S$.
– Federico
2 days ago
What do you mean? $M^S={f:M to S}$ is a well defined set for every $M$ and $S$.
– Federico
2 days ago
Regarding charts, yes, I am aware of them being maps from a manifold M to Rd, but not to R. Nevertheless, it still confuses me. How exactly do you chart a manifold?
– Žarko Tomičić
2 days ago
Regarding charts, yes, I am aware of them being maps from a manifold M to Rd, but not to R. Nevertheless, it still confuses me. How exactly do you chart a manifold?
– Žarko Tomičić
2 days ago
There can be plenty of ways to describe any specific function.
– Federico
2 days ago
There can be plenty of ways to describe any specific function.
– Federico
2 days ago
For instance, $S^2subsetmathbb R^3$ can be endowed with a manifold structure. But since it is also already embedded inside $mathbb R^3$, you can also just take any function defined on the ambient space and restrict it.
– Federico
2 days ago
For instance, $S^2subsetmathbb R^3$ can be endowed with a manifold structure. But since it is also already embedded inside $mathbb R^3$, you can also just take any function defined on the ambient space and restrict it.
– Federico
2 days ago
|
show 12 more comments
Žarko Tomičić is a new contributor. Be nice, and check out our Code of Conduct.
Žarko Tomičić is a new contributor. Be nice, and check out our Code of Conduct.
Žarko Tomičić is a new contributor. Be nice, and check out our Code of Conduct.
Žarko Tomičić is a new contributor. Be nice, and check out our Code of Conduct.
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What is a definition of function?
– Anubhav Mukherjee
2 days ago
@Anubhav Mukherjee Map from manifold to R.
– Žarko Tomičić
2 days ago