Tangent space

One can attach to every point x of a differentiable manifold a tangent space, a real vector space that intuitively contains the possible “directions” at which one can tangentially pass through x. The elements of the tangent space are called tangent vectors atx. All tangent spaces of a connected manifold have the same dimension, equal to the dimension of the manifold

Definition as velocities of curves

Suppose M is a C^k manifold (k\geq 1) and x is a point in M. Pick a chart \varphi:U\to\mathbf{R}^n where U is an open subset of M containing x. Suppose two  curves \gamma_1:(-1,1)\to M and \gamma_2:(-1,1)\to M with \gamma_1(0)=\gamma_2(0)=x are given such that \varphi\circ\gamma_1 and \varphi\circ\gamma_2 are both differentiable at 0. Then \gamma_1 and \gamma_2 are called equivalent at 0 if the ordinary derivatives of \varphi\circ\gamma_1 and \varphi\circ\gamma_2 coincide. This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vector of M at x. The equivalence class of the curve \gamma is written as \gamma'(0). The tangent space of M at x, denoted by T_xM, is defined as the set of all tangent vectors, it does not depened on the choice of chart \varphi.

To define the vector space operations on T_xM, we use a chart \varphi:U\to\mathbf{R}^n and define the map d\varphi_x(\gamma'(0))=\frac{d}{dt}(\varphi\circ\gamma)(0). It turns out that this map is bijective and can thus be used to transfer the vector space operation from \mathbf{R}^n over T_xM, turning the latter into an n-dimensional real vector space. Again, one need to check that this construction does not depend on the particular char \varphi chosen, and in fact is does not.

Definition via derivations

Suppose that M is a C^\infty manifold. A real valued function f:M\to\mathbf{R} belongs to C^\infty(M) if f\circ\varphi^{-1} is infinitely differentaible for every chart \varphi:U\to \mathbf{R}. C^\infty(M) is a real associative algebra for the pointwise product and sum of functions and scalar multiplication.

Pick a point x in M. A derivation at x is a liner map D:C^\infty(M)\to\mathbf{R} that has the property that for all f,g in C^\infty(M):

\displaystyle D(fg)=D(f)\times  g(x)+f(x)\times D(g)

modeled on the product rule of calculus. If we define addition and scalar multiplication for such derivations by

\displaystyle (D_1+D_2)(f)=D_1(f)+D_2(f)\ \text{and}\ (\lambda D)(f)=\lambda D(f)

we get a real vector space which we define as the tangent space T_xM.

The relation between the  tangent vectors defined earlier and derivatoin is as follows: if \gamma is a curve with tangent vector \gamma'(0), then the corresponding derivation D(f)=(f\circ\gamma)'(0).

\displaystyle \gamma'(0)\mapsto D_\gamma \  \text{where}\ D_\gamma(f)=\frac{d}{dt}(f\circ \gamma)\Big|_{t=0}.