mathematics-physics-wiki/docs/en/mathematics/differential-geometry/differential-manifolds.md

Differential manifolds

In the following sections we make use of the Einstein summation convention introduced in vector analysis and \mathbb{K} = \mathbb{R} or \mathbb{K} = \mathbb{C}.

Definition

Differential geometry is concerned with differential manifolds, smooth continua that are locally Euclidean.

Definition 1: let n \in \mathbb{N}, a $n$-dimensional differential manifold is a Hausdorff (T2) space M furnished with a family of smooth diffeomorphisms \phi_\alpha: \mathscr{D}(\phi_\alpha) \to \mathscr{R}(\phi_\alpha) with \mathscr{D}(\phi_\alpha) \subset\mathrm{M} and \mathscr{R}(\phi_\alpha) \subset E, with the following axioms

1. \mathscr{D}(\phi_\alpha) is open and \bigcup_{\alpha \in \mathbb{N}} \mathscr{D}(\phi_\alpha) =\mathrm{M},
2. if \Omega = \mathscr{D}(\phi_\alpha) \cap \mathscr{D}(\phi_\beta) \neq \empty then \phi_\alpha(\Omega), \phi_\beta(\Omega) \subset E are open sets and \phi_\alpha \circ \phi_\beta^{-1}, \phi_\beta \circ \phi_\alpha are diffeomorphisms,
3. the atlas \mathscr{A} = \{(\mathscr{D}(\phi_\alpha), \phi_\alpha)\} is maximal.

with E a $n$-dimensional Euclidean space.

The last axiom ensures that any chart is tacitly assumed to be already contained in the atlas.

Transformations

Definition 2: let p,q \in \mathrm{M} be points on the differential manifold and let \psi: \mathscr{D}(\psi) \to\mathrm{M}: p \mapsto \psi(p) \overset{\text{def}}{=} q be a transformation on the manifold, we define two diffeomorphisms

\phi_\alpha: \mathscr{D}(\phi_\alpha) \to \mathscr{R}(\phi_\alpha): p \mapsto \phi_\alpha(p) \overset{\text{def}}{=} x,

\phi_\beta: \mathscr{D}(\phi_\beta) \to \mathscr{R}(\phi_\beta): q \mapsto \phi_\beta(q) \overset{\text{def}}{=} y,

with \mathscr{D}(\phi_{\alpha,\beta}) \subset\mathrm{M} and \mathscr{R}(\phi_{\alpha,\beta}) \subset E. Then we have a coordinate transformation given by

\phi_{\alpha \beta}^\psi = \phi_\beta \circ \psi \circ \phi_\alpha^{-1}: x \mapsto y,

then \phi_{\alpha \beta}^\psi is an active transformation if p \neq q and \phi_{\alpha \beta}^\psi is a passive transformation if p = q.

To clarify the definitions, a passive transformation corresponds only to a descriptive transformation. Whereas an active transformation corresponds to a transformation on the manifold M.

A passive transformation may also be given directly by \phi_\beta \circ \phi_\alpha: x \mapsto y since \psi = \mathrm{id} in this case. Note that the definitions could also have been given by the inverse as the transformations are all diffeomorphisms.

Fiber bundles

(This subsection should probably be moved to a more general setting of manifolds.)

Definition 3: a fiber V_x at a point x \in \mathrm{M} on a manifold is a finite dimensional vector space. With the collection of fibers V_x for all x \in \mathrm{M} define the fiber bundle as

V = \bigcup_{x \in \mathrm{M}} V_x.

Then by definition we have the projection map \pi given by

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

and its inverse

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

Similarly, a dual fiber V_x^* may be defined for x \in \mathrm{M}, with its fiber bundle defined by

V^* = \bigcup_{x \in \mathrm{M}} V_x^*.

Definition 4: a tensor fiber \mathscr{B}_x at a point x \in \mathrm{M} on a manifold is defined as

\mathscr{B}x = \bigcup{p,q \in \mathbb{N}} \mathscr{T}^p_q(V_x).

With the collection of tensor fibers \mathscr{B}_x for all x \in \mathrm{M} define the tensor fiber bundle as

\mathscr{B} = \bigcup_{x \in \mathrm{M}} \mathscr{B}_x.

Then for a point x \in \mathrm{M} we have a tensor \mathbf{T} \in \mathscr{B}_x such that

\mathbf{T} = T^{ij}_k \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k,

with T^{ij}_k \in \mathbb{K} holors of \mathbf{T}. Furthermore, \{\mathbf{e}_i\}_{i=1}^n a basis of V_x and \{\mathbf{\hat e}^i\}_{i=1}^n a basis of V_x^*.

Definition 5: a tensor field \mathbf{T} on a manifold M is a section

\mathbf{T} \in \Gamma(\mathrm{M}, \mathscr{B}),

of the tensor fiber bundle \mathscr{B}.

Therefore, a tensor field assigns a tensor fiber (or tensor) to each point on a section of the manifold. These tensors may vary smoothly along the section of the manifold.

Tangent bundles

Definition 6: 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.

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.

Definition 7: 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 vector 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 8: 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 bundles

Definition 9: 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.

Push forward and pull back

Definition 10: let \mathrm{M}, \mathrm{N} be two differential manifolds with \dim \mathrm{N} \geq \dim \mathrm{M} and let \psi: \mathrm{M} \to \mathrm{N} be the diffeomorphism between the manifolds. Then we define the pull back \psi^* and push forward \psi_* operators, such that for \mathbf{v} \in \mathrm{T}_x \mathrm{M} and \bm{\omega} \in \mathrm{T}_{\psi(x)}^* \mathrm{M} we have

\mathbf{k}x(\psi^* \bm{\omega}, \mathbf{v}) = \mathbf{k}{\psi(x)}(\bm{\omega}, \psi_* \mathbf{v}),

for all x \in \mathrm{M}.

Which indicates the proper separation between the elements of both spaces.