Intrinsic definition on a manifold

[kiuhnm]

I'm reading "The Geometry of Physics" by Frankel. Exercise 1.3(1) asks what would be wrong in defining $||X||$ in an $M^n$ by
$$||X||^2 = \sum_j (X_U^j)^2$$ The only problem I can see is that that definition is not independent of the chosen coordinate systems and thus not intrinsic to $M^n$. Is there anything else I'm missing?

BTW, I'm not sure this is Differential Geometry... Is this Topology?

 

[Orodruin]

The only problem I can see is that that definition is not independent of the chosen coordinate systems and thus not intrinsic to MnMnM^n. Is there anything else I'm missing?

Is that not enough?

 

[kiuhnm]

Is that not enough?

Don't get mad. I just wanted to be sure. I'm learning this stuff on my own so I can have doubts from time to time :)

 

[fresh_42]

I'm reading "The Geometry of Physics" by Frankel. Exercise 1.3(1) asks what would be wrong in defining $||X||$ in an $M^n$ by
$$||X||^2 = \sum_j (X_U^j)^2$$ The only problem I can see is that that definition is not independent of the chosen coordinate systems and thus not intrinsic to $M^n$. Is there anything else I'm missing?

BTW, I'm not sure this is Differential Geometry... Is this Topology?

It's always a bit of both, but as we measure lengths, it is more geometric than topological.

And, yes, as there is no natural way to choose a chart, we want to have a definition which is independent of the chart. This is always the basic principle: Pull it down into the reals (or complex), do what has to be done, and lift it up again. This way we stay as general as possible on the choices of manifolds, but are still able to do calculus and geometry. The costs are: it can only locally be done and the difficulties will start, if we want to compare two different local events, e.g. tangents.

 

[Orodruin]

Don't get mad.

I'm not mad, it was an honest question.

 

[kiuhnm]

I'm not mad, it was an honest question.

I was joking. I interpreted your reply as "That's more than enough".

 

[kiuhnm]

It's always a bit of both, but as we measure lengths, it is more geometric than topological.

And, yes, as there is no natural way to choose a chart, we want to have a definition which is independent of the chart. This is always the basic principle: Pull it down into the reals (or complex), do what has to be done, and lift it up again. This way we stay as general as possible on the choices of manifolds, but are still able to do calculus and geometry. The costs are: it can only locally be done and the difficulties will start, if we want to compare two different local events, e.g. tangents.

I'm still reading chapter 1 of that book. For now I know that the (tangent) vectors are all the vectors that transform as $$X = \frac{\partial x}{\partial y} Y,$$ where $X$ and $Y$ are the same vector expressed in the $(x_i)$ and $(y_i)$ coordinate systems, respectively. That's only required (and makes sense) when two patches $(U,x)$ and $(V,y)$ overlap, of course. That's what I understood.

 

[fresh_42]

I'm still reading chapter 1 of that book. For now I know that the (tangent) vectors are all the vectors that transform as $$X = \frac{\partial x}{\partial y} Y,$$ where $X$ and $Y$ are the same vector expressed in the $(x_i)$ and $(y_i)$ coordinate systems, respectively. That's only required (and makes sense) when two patches $(U,x)$ and $(V,y)$ overlap, of course. That's what I understood.

Yes, but there is a difference between
$$X = \frac{\partial x}{\partial y} Y \text{ and }X_p = \left.\frac{\partial x}{\partial y}\right|_p Y$$
The first is a vector field $\{\,(p,X_p)\,|\,p\in M\,\}$ and the second a tangent space $X_p$. Now if we have two tangents $X_p$ and $X_q$ we cannot compare them automatically, as $p$ and $q$ might not be covered by the same charts. That's were the consequences of "locally Euclidean" comes into play. There is no "global" anymore. Imagine our manifold is the surface of Mars. Then all we have are the charts given by some orbiters. We cannot simply walk from one point to the next - we have to attach our charts.

 

[WWGD]

Yes, but there is a difference between
$$X = \frac{\partial x}{\partial y} Y \text{ and }X_p = \left.\frac{\partial x}{\partial y}\right|_p Y$$
The first is a vector field $\{\,(p,X_p)\,|\,p\in M\,\}$ and the second a tangent space $X_p$. Now if we have two tangents $X_p$ and $X_q$ .

Tangent vectors $X_p, X_q$?

 

[kiuhnm]

Yes, but there is a difference between
$$X = \frac{\partial x}{\partial y} Y \text{ and }X_p = \left.\frac{\partial x}{\partial y}\right|_p Y$$
The first is a vector field $\{\,(p,X_p)\,|\,p\in M\,\}$ and the second a tangent space $X_p$. Now if we have two tangents $X_p$ and $X_q$ we cannot compare them automatically, as $p$ and $q$ might not be covered by the same charts. That's were the consequences of "locally Euclidean" comes into play. There is no "global" anymore. Imagine our manifold is the surface of Mars. Then all we have are the charts given by some orbiters. We cannot simply walk from one point to the next - we have to attach our charts.

I was being sloppy (just like the book) and fixing $p$, so my $X$ was really a $X_p$. In my notation, $X_p$ is a (tangent) vector and $M^n_p$ is the tangent space, if $M^n$ is the $n$-dimensional manifold.