Curves and tangent vectors in a manifold setting

[CAF123]

Consider the following definition: ($M$ denotes a manifold structure, $U$ are subsets of the manifold and $\phi$ the transition functions)

Def: A smooth curve in $M$ is a map $\gamma: I \rightarrow M,$ where $I \subset \mathbb{R}$ is an open interval, such that for any chart $(U,\phi)$, the map $\phi \circ \gamma : I \rightarrow \mathbb{R}^n$ is smooth.

My first question is, why do we define a smooth curve in this way? In particular, why is the map $\phi \circ \gamma$ a good object to consider? The only thing that comes to mind is that now we have a function defined from $\mathbb{R} \rightarrow \mathbb{R}^n$ so differentiation is well defined and thus one may introduce the concept of a tangent vector (as below).

Now let $f: M \rightarrow \mathbb{R}$ be a smooth function on $M$ and $\gamma: I \rightarrow M$, smooth curve as before. Then $f \circ \gamma : I \rightarrow \mathbb{R}$ is smooth. Hence we take a derivative to find the rate of change of $f$ along the curve $\gamma$: $$\frac{d}{dt}f(\gamma(t)) = [(f \circ \phi^{-1}) \circ (\phi \circ \gamma)]'(t) = \sum_{i=1}^n \left(\frac{\partial (f \circ \phi^{-1})}{\partial x^i}\right)_{\phi(\gamma(t))} \frac{d}{dt} x^i(\gamma(t))$$

My next question is to simply understand how this equation comes about. I can see it is some application of the chain rule but I am struggling with the precise details of the equation, mostly in how the final equality comes about and the subscript on the $\partial (f \circ \phi^{-1})/\partial x^i$ term. Many thanks!

 

[Orodruin]

My first question is, why do we define a smooth curve in this way?

It allows us to do essentially everything on a smooth curve on a manifold that you can do with a smooth curve in Rⁿ. (You can also check that the definition is coordinate independent.) To quote one of my undergrad teachers (linear algebra, not differential geometry): "It is not silly, it is a definition. It may be a silly definition, but you simply have to live with it."

My next question is to simply understand how this equation comes about. I can see it is some application of the chain rule but I am struggling with the precise details of the equation, mostly in how the final equality comes about and the subscript on the $\partial (f \circ \phi^{-1})/\partial x^i$ term. Many thanks!

$f\circ \phi^{-1}$ is a function from the coordinate chart $U$ to $\mathbb R$ and it therefore has a dependence on the coordinates $x^i$, which is what you differentiate this function with respect to. The full function you have is a composition of a function from $\mathbb R$ to $\mathbb R^n$ and a function from $\mathbb R^n$ to $\mathbb R$ and you therefore can apply the chain rule ($x^i(\gamma(t))$ are simply the coordinate functions evaluated at the curve for some $t$).

 

[Fredrik]

Consider the following definition: ($M$ denotes a manifold structure, $U$ are subsets of the manifold and $\phi$ the transition functions)

Def: A smooth curve in $M$ is a map $\gamma: I \rightarrow M,$ where $I \subset \mathbb{R}$ is an open interval, such that for any chart $(U,\phi)$, the map $\phi \circ \gamma : I \rightarrow \mathbb{R}^n$ is smooth.

My first question is, why do we define a smooth curve in this way? In particular, why is the map $\phi \circ \gamma$ a good object to consider? The only thing that comes to mind is that now we have a function defined from $\mathbb{R} \rightarrow \mathbb{R}^n$ so differentiation is well defined and thus one may introduce the concept of a tangent vector (as below).

Smoothness of functions between subsets of $\mathbb R^n$ is defined as "differentiable infinitely many times". When we define smoothness of functions between manifolds, we want the new kind of smoothness to be related to the old. Since a manifold is covered by the domains of a bunch of smoothly compatible coordinate systems (such that $x\circ y^{-1}$ is smooth for all coordinate systems x,y), it's very natural to define $f:M\to N$ to be smooth when $x\circ f\circ y^{-1}$ is smooth for all coordinate systems x on N and y on M.

Now let $f: M \rightarrow \mathbb{R}$ be a smooth function on $M$ and $\gamma: I \rightarrow M$, smooth curve as before. Then $f \circ \gamma : I \rightarrow \mathbb{R}$ is smooth. Hence we take a derivative to find the rate of change of $f$ along the curve $\gamma$: $$\frac{d}{dt}f(\gamma(t)) = [(f \circ \phi^{-1}) \circ (\phi \circ \gamma)]'(t) = \sum_{i=1}^n \left(\frac{\partial (f \circ \phi^{-1})}{\partial x^i}\right)_{\phi(\gamma(t))} \frac{d}{dt} x^i(\gamma(t))$$

My next question is to simply understand how this equation comes about. I can see it is some application of the chain rule but I am struggling with the precise details of the equation, mostly in how the final equality comes about and the subscript on the $\partial (f \circ \phi^{-1})/\partial x^i$ term. Many thanks!

The chain rule can be written $(f\circ g)_{,i}(x)=f_{,j}(g(x)) g^j{}_{,i}(x)$ when $f$ is real-valued. If the domain of $g$ is a subset of $\mathbb R$, we can simplify this to $(f\circ g)'(x)=f_{,j}(g(x))(g^j)'(x)$.