Let’s start with a differentiable manifold M of dimension . Throughout this topic, we denote by a point on and its local chart (at ). A point is determined by hence it is often identified with . We usually denote by the local coordinates of .
Definition 1. A tangent vector at is a map defined on the set of the differentiable functions in a neighborhood of , where satisfies the following conditions
- is linear, that is to say: if , then .
- if is flat at , i.e. at .
Definition 2. The tangent space at is the set of tangent vectors at .
From the definition 1, let us show that the tangent space of definition 2 has a natural vector space structure of dimension . We set
With this sum and this product, is a vector space. And now let us exhibit a basis. It is reasonable to define the tangent vector at . Precisely,
Definition 3. The tangent vector at is defined to be
The vectors are independent and they form a basis for . We usually call the directional derivative of in the direction . For an arbitrary vector , one can define the directional derivative of in the direction as following
where denotes the -th component of in this coordinate chart.
where denotes the inner product of tangent vectors at defined by the metric .
We now express the local form of the gradient at . By definition 4, one has in the local coordinates
If we assume we then have
which implies after multiplying both sides by the matrix
We end this topic by showing what is? Roughly speaking, at a point since is a vector, then is nothing but its magnitude. To be exact, one defines
In Riemannian geometry, the lower index means differentiation and the upper index means component, therefore, we usually use to denote the quantity . With this notation, .