Tangent space at a point

This article defines a basic construct that makes sense on any differential manifold
View a complete list of basic constructs on differential manifolds

Definition

Let $M$ be a differential manifold and $p\in M$. The tangent space at $p$ to $M$, denoted $T_{p}(M)$, is defined in a number of equivalent ways.

In the language of derivations

Consider the algebra $C^{\infty }(M)$ of infinitely differentiable functions on $M$. Then, the tangent space $T_{p}(M)$ is defined as the vector space of all derivations:

$d:C^{\infty }(M)\to \mathbb {R}$

By $d$ being a derivation, we mean:

• $d$ is a $\mathbb {R}$-linear map
• Given functions $f,g\in C^{\infty }(M)$ we have:

$d(fg)=(df)(g(p))+f(p)(dg)$

In the language of curves

The tangent space $T_{p}(M)$ is defined, as a set, as the quotient of the set of all curves $\gamma :(-1,1)\to M$ where $\gamma (0)=p$, by the following equivalence relation: $\gamma _{1}=\gamma _{2}$ if, under the mapping to an open subset in $\mathbb {R} ^{n}$ via a coordinate chart, $\gamma _{1}'(0)=\gamma _{2}'(0)$.

With this definition, addition is not very clear.

In the language of local coordinate charts

In this language, the tangent space is defined as the tangent space to its image in $\mathbb {R} ^{n}$ under a local coordinate chart. Fill this in later