Notation for vector tangent to a curve on a differentiable manifold

Question

A result for differentiable maps that I'll use: let $F:M\to N$, where $M,N$ are smooth manifolds and $F$ is a smooth map. If $(U,x)$ is a chart around $p\in M$ and $(V,y)$ is a chart around $F(p)\in N$, then $$d_p(F)\bigg(\frac{\partial}{\partial x^j}\bigg|_p\bigg)=\bigg(\frac{\partial}{\partial x^j}\bigg|_p\bigg)(y^i\circ F)\frac{\partial}{\partial y^i}\bigg|_{F(p)}\equiv\frac{\partial(y^i\circ F)}{\partial x^j}(p)\frac{\partial}{\partial y^i}\bigg|_{F(p)}$$

Now let $\lambda:(a,b)\to M$ be some curve from interval $(a,b)$ to $M$. $(a,b)$ itself can be taken to be a 1D smooth manifold. So I can use the result above like so: $$d_{t_0}(\lambda)\bigg(\frac{d}{dt}\bigg|_{t_0}\bigg)=\bigg(\frac{d}{dt}\bigg|_{t_0}\bigg)(x^i\circ \lambda)\frac{\partial}{\partial x^i}\bigg|_{\lambda(t_0)}\equiv\frac{d(x^i\circ \lambda)}{dt}(t_0)\frac{\partial}{\partial x^i}\bigg|_{\lambda(t_0)}$$

But a book I'm reading calls the expression on the right as $\frac{d\lambda}{dt}(t_0)$. What's the reasoning for getting from the RHS of the above equation to $\frac{d\lambda}{dt}(t_0)$?

Answer 1

I don't think one can provide a purely mathematical answer, in as much it is a notational convention. Curves, and their derivatives, play an important role in differential geometry and you want to get a shorthand for what you write above (the tangent map of $\lambda$ acting on $d/dt$).

I don't know what book you are using, but for instance you will find this remark in TU's book, page 92:

When $c: ]a,b[\to \mathbb{R}$ is a curve with target space $\mathbb{R}$, the notation $c′(t)$ can be a source of confusion. Here $t$ is the standard coordinate on the domain $]a,b[$. Let $x$ be the standard coordinate on the target space $\mathbb{R}$. By our definition, $c′(t)$ is a tangent vector at $c(t)$, hence a multiple of $d/dx|_c(t)$. On the other hand, in calculus notation $c′(t)$ is the derivative of a real-valued function and is therefore a scalar. If it is necessary to distinguish between these two meanings of $c′(t)$ when $c$ maps into $\mathbb{R}$, we will write $\dot c(t)$ for the calculus derivative.

By the way, TU defines $c'(t_0)=c_*(d/dt|_{t_0})$, and he mentions explicitely that an alternative notation is $(dc/dt)(t_0)$ (which would agree with your book's notation).

Answer 2

This is just a definition. Note that the expression $\frac{d\lambda}{dt}(t_0)$ makes no sense for abstract manifolds a priori.
The reason for setting

$$\frac{d\lambda}{dt}(t_0)=d_{t_0}(\lambda)\left(\left.\frac{d}{dt}\right|_{t_0}\right)$$

is due to the fact that it generalizes the usual case in which $\lambda:(a,b)\to \mathbb{R}^n$ and

$$\frac{d\lambda}{dt}(t_0)=D\lambda(t_0)\,\cdot 1$$

to a setting of abstract manifolds.