Some basic definitions

Definition 1. A tangent vector $V$ at $p\in S$ is an equivalence class of curves $\gamma:[a,b]\to S$ such that $\gamma(a)=p$, where two curves $\gamma_1$ and $\gamma_2$ are equivalent if their derivatives at $t=a$ are equal, i.e., $\gamma_1'(a)=\gamma_2'(a)$. Also, it can be defined as an operator from $C^{\infty}(S)$ to $\mathbb R$

$$ V(f)=(f\circ \gamma)'(0) $$

Definition 2. The tangent vector space $T_p S$ is the set of all tangent vectors at $p$. It is a vector space, with the addition and scalar multiplication of tangent vectors defined pointwise, and zero vector given by the equivalence class of the constant curve at $p$.

Definition 3. The differential of a map $F:S\to \mathbb R^m$ at a point $p\in S$ is a linear map $dF_p:T_p S\to \mathbb R^m$ defined as follows: for any tangent vector $V\in T_p S$, we define $dF_p(V)$ as a vector whose action on a function $g$ in $C^{\infty}(\mathbb R^m$ is

$$ dF_p(V)(g)=V(g\circ F) $$

(observe that $g\circ F \in C^{\infty}(S)$).

With all this three ingredients, you have that $dF_p$ is linear since

$$ dF_p(V+W)(g)=(V+W)(g\circ F)=V(g\circ F)+W(g\circ F)= $$ $$ =dF_p(V)(g)+dF_p(W)(g) $$

and

$$ dF_p(\lambda V)(g)=\lambda V(g\circ F)=\lambda dF_p(V)(g) $$

Steps on the abstract definition of tangent vector:

It can be shown that in $\mathbb{R}^n$ traditional vectors are the same as derivations of (the germ of) functions.

We define vectors at $p$ in a manifold $M$ as derivations of the germ of functions at $p$, inspired in the previous item.

The collection of all of then constitute the tangent space at $p$: $T_p M$.

Given a smooth curve $\gamma$ such that $\gamma(0)=p$, we can define a tangent vector at $p$ (in the sense above) and denote it by $\gamma'(0)$ in the following way

$$ \gamma'(0) (f)=\frac{d}{dt}(f \circ \gamma)(0) $$

If we take as particular $\gamma$ the coordinate curves of a specific chart ($x_1, x_2, \ldots, x_n$) we obtain $n$ vectors that constitutes a basis of $T_p M$, and we will denote them by $\partial_{x_1}, \partial_{x_2}, \ldots, \partial_{x_n}$.

Then, we can show the converse: every tangent vector at $p$ arises as the tangent vector of a smooth curve $\gamma$ (not unique). We have, now, two characterizations of tangent vectors.

It there exists a third characterization: as maps from $C(p)$ to $\mathbb{R}^n$, where $C(p)$ is the collection of all coordinate charts whose domain contains $p$. Obviously, the map has to verify some conditions (contravariant change).

Even more, if $M$ is immersed in $\mathbb{R}^{N}$ then we could characterize tangent vectors as the usual vectors of $\mathbb{R}^{N}$. This would be the {\bf fourth} way of think about tangent vectors.

Important properties: the canonical form of a regular vector field,the canonical form of commuting vector fields and the flow theorem for vector fields.

________________________________________

Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

INDEX: