mathematics-physics-wiki/docs/en/mathematics/differential-geometry/tangent-spaces.md

Tangent spaces

Let \mathrm{M} be a differential manifold with \dim \mathrm{M} = n \in \mathbb{N} used throughout the section.

Definition

Definition 1: let f \in C^{\infty}(\mathrm{M}) with C^{\infty} the class of smooth functions and M a differential manifold. A derivation of f at x \in \mathrm{M} is defined as a linear map \mathbf{v}_x: C^\infty(\mathrm{M}) \to \mathbb{K} that satisfies

\forall f,g \in C^{\infty}(\mathrm{M}): \mathbf{v}_x(f g) = (\mathbf{v}_xf) g + f (\mathbf{v}_x g).

Let \mathrm{T}_x\mathrm{M} be the set of all derivations at x such that \mathbf{v}_x \in \mathrm{T}_x\mathrm{M}. With \mathrm{T}_x\mathrm{M} denoted as the tangent space at x.

We may think of the tangent space at a point x \in \mathrm{M} as a space attached to x on the differential manifold M.

Properties of tangent spaces

Theorem 1: let M be a differential manifold and let x \in \mathrm{M}, the tangent space \mathrm{T}_x\mathrm{M} is a vector space.

??? note "Proof:"

Will be added later.

Thus, the tangent space is a vector space attached to x \in \mathrm{M} on the differential manifold. It follows that its vectors have interesting properties.

Theorem 2: let M be a differential manifold, let x \in \mathrm{M} and let \mathbf{v}_x \in \mathrm{T}_x\mathrm{M}, then we have that

\forall f \in C^{\infty}(\mathrm{M}): \mathbf{v}_x f = v^i \partial_i f(x),

such that \mathbf{v}_x = v^i \partial_i \in \mathrm{T}_x\mathrm{M} is denoted as a tangent vector in the tangent space \mathrm{T}_x\mathrm{M}.

??? note "Proof:"

Will be added later.

Theorem 2 adds the notion of tangent vectors to the explanation of the tangent space. The tangent space at a point on the manifold thus represents the space of tangent vectors.

Proposition 1: let M be a differential manifold of \dim\mathrm{M} = n \in \mathbb{N}. The tangent space \mathrm{T}_x\mathrm{M} has dimension n such that

\forall x \in \mathrm{M}: \dim \mathrm{T}_x\mathrm{M} = \dim\mathrm{M}

and is span by the vector basis \{\partial_i\}_{i=1}^n.

??? note "Proof:"

Will be added later.

Proposition 1 states that the tangent space is of the same dimension as the manifold and its basis are partial derivative operators. In the context of the covariant basis, this definition of the basis leaves out the coordinate map, but is in fact equivalent to the covariant basis.

As a last step in the explanation, we may think of the 2 dimensional surface of a sphere, which may define a differential manifold M. The tangent space at a point x \in \mathrm{M} on the surface of the sphere may then be compared to the tangent plane to the sphere attached at point x \in \mathrm{M}. The catch is that the 3 dimensional space necessary to understand this construction exists only in our imagination and not in the mathematical construct.

Tangent bundle

Definition 2: let M be a differential manifold, the collection of tangent spaces \mathrm{T}_x\mathrm{M} for all x \in \mathrm{M} define the tangent bundle as

\mathrm{TM} = \bigcup_{x \in \mathrm{M}} \mathrm{T}_x\mathrm{M}.

In particular, we may think of the tangent bundle \mathrm{TM} as a subspace \mathrm{TM} \subset V of the fiber bundle V for a differential manifold. With the special properties given in theorem 2 and proposition 1.

The connection of each tangent vector to its base point may be formalised with the projection map \pi which in this case is given by

\pi: \mathrm{TM} \to\mathrm{M}: (x, \mathbf{v}) \mapsto \pi(x, \mathbf{v}) \overset{\text{def}}{=} x,

and its inverse

\pi^{-1}:\mathrm{M} \to \mathrm{TM}: x \mapsto \pi^{-1}(x) \overset{\text{def}}{=} \mathrm{T}_x\mathrm{M}.

Definition 3: a vector field \mathbf{v} on a differential manifold M is a section

\mathbf{v} \in \Gamma(\mathrm{TM}),

of the tangent bundle \mathrm{TM}.

Cotangent spaces

Definition 4: let M be a differential manifold and \mathrm{T}_x\mathrm{M} the tangent space at x \in \mathrm{M}. We define the cotangent space \mathrm{T}_x^*\mathrm{M} as the dual space of \mathrm{T}_x\mathrm{M}

\mathrm{T}_x^\mathrm{M} = (\mathrm{T}_x\mathrm{M})^.

Then every element \bm{\omega}_x \in \mathrm{T}_x^*\mathrm{M} is a linear map \bm{\omega}_x: \mathrm{T}_x\mathrm{M} \to \mathbb{K} denoted as the cotangent vector.

This definition is a logical consequence of the notion of the dual vector space. It then also follows that the dual cotangent space is isomorphic to the tangent space at a point x \in \mathrm{M}.

Theorem 3: let \mathrm{M} be a differential manifold of \dim \mathrm{M} = n \in \mathbb{N}, then we have that for every x \in \mathrm{M} the basis \{dx^i\}_{i=1}^n of \mathrm{T}_x^*\mathrm{M} is uniquely determined by

dx^i(\partial_j) = \delta^i_j,

for each basis \{\partial_j\}_{j=1}^n in \mathrm{T}_x\mathrm{M}.

??? note "Proof:"

The proof follows directly from theorem 1 in [dual vector spaces](). 

The choice of dx^i can be explained by taking the differential df = \partial_i f dx^i \in \mathrm{T}_x^*\mathrm{M} with f \in C^\infty(\mathrm{M}). Then if we take

\mathbf{k}_x(df, \mathbf{v}) = \mathbf{k}(\partial_i f dx^i, v^j \partial_j) = v^j \partial_i f \mathbf{k}(dx^i, \partial_j) = v^j \partial_i f \delta^i_j = v^i \partial_i f = \mathbf{v} f,

with \mathbf{k}_x: \mathrm{T}_x^*\mathrm{M} \times \mathrm{T}_x\mathrm{M} \to \mathbb{K} the Kronecker tensor at x \in \mathrm{M}. Which shows that defining the basis of the cotangent space as differentials corresponds with respect to the basis of the tangent space.

So, a cotangent vector \bm{\omega}_x \in \mathrm{T}_x^*\mathrm{M} may be decomposed into

\bm{\omega}_x = \omega_i dx^i.

In the context of the contravariant basis, this definition of the basis leaves out the coordinate map, but is in fact equivalent to the contravariant basis.

Cotangent bundle

Definition 5: let M be a differential manifold, the collection of cotangent spaces \mathrm{T}_x^*\mathrm{M} for all x \in \mathrm{M} define the cotangent bundle as

\mathrm{T^M} = \bigcup_{x \in \mathrm{M}} \mathrm{T}_x^\mathrm{M}.

Thus, we may think of the cotangent bundle \mathrm{T^*M} as a subspace \mathrm{T^*M} \subset V^* of the dual fiber bundle V^* for a differential manifold.