4.6 Differential Geometry

Having carefully prepared the foundations which present the scenery for the geometrical considerations at hand, it is time to turn to geometric entities. It is again prudent to present a very simple case first, which allows to build higher structures upon.

Definition 45 (Curve) A smooth map of an interval I ⊂ ℝ  to a differentiable manifold 𝔐  (Definition 37) is called a curve:

γ : I ⊂ ℝ →  𝔐.                               (4.63)

Two curves are not identical, in the strictest of senses, if their images coincide, while their parametrization differ.

Using a curve, every number s ∈ I is mapped to a point of 𝔐  , which may be used as input for the charts of the manifold (Definition 35), thereby effectively creating the new mapping:

κ ∘ γ : I → ℝn                                        (4 .64a)
x (s) : I → ℝ,∀i ∈ [0,...,n − 1]                      (4.64b)

This pullback (Definition 44) by the charts also results in coordinate representations for the involved points. While the properties of a considered curve do not depend on the choice of a particular coordinate system, the mere existence of the mapping allows to explain the concept of a differentiable curve.

Definition 46 (Differentiable curve) A curve γ(s)  is considered to be differentiable, if the functions describing its coordinate representations xi(s)  as defined by Equation 4.64 are differentiable functions.

Having defined differentiable curves, it is only fitting to recall basic mathematics and how tangents to graphs of simple functions may be linked to differentials, and extend the concept by applying similar reasoning in this setting.

Definition 47 (Tangent curves)

Two curves γ1(t),γ2(t)  are tangent to each other in a point P , if

   P =  γ1(t0) = γ2(t0)                           (4.65)
 d           d
---γ1(t)|t0 = --γ2(t)|t0                           (4.66)
dt           dt

The differentiation is with respect to a single parameter and touches the coordinate expressions according to Equations 4.64.


Figure 4.3: Several curves tangent in a point P .

Figure 4.3 illustrates that Definition 47 can be met by several, in fact infinitely many, curves. However, recalling Definition 9 it is possible to extract essential parts of information which leads to the following definition introducing a key geometric entity.

Definition 48 (Tangent vector space) Considering the set of all curves passing through a point P∈𝔐 , the identity of the first derivatives defines a set of equivalence classes (Definition 9):

˙γ = [γ ]                                   (4 .67)
The equivalence classes at a given point P define a space TM
    P  , which carries the structure of a vector space (Definition 16) and tangent to 𝔐  at the point P . The dimensions of the manifold 𝔐  and TMP are identical
dim (𝔐 ) = dim (T MP  ).                           (4.68)

The vectors now available after Definition 48 alone do not suffice to easily develop complex models of physical processes. A method of evaluation, a measurement, of the vectors is also required, which motivates the next definition.

Definition 49 (Dual vector space) Given a vector space 𝒱 , its dual space 𝒱∗ is comprised by the linear functionals

α ∈ 𝒱 ∗ : 𝒱 → ℝ.                              (4.69)

The dual space  ∗
𝒱 also carries the structure of a vector space (Definition 16), its elements are called co-vectors or one-forms. In particular it is possible to define a non-degenerate bilinearform, called the scalar product.

⟨α, a⟩ : 𝒱∗ × 𝒱 → ℝ                             (4.70)
Where non-degenerate means that ⟨α, a⟩ = 0  may only occur, if either
α =  0    ∀a  ∈ 𝒱   or                           (4.71)
 a = 0    ∀ α ∈ 𝒱 ∗.                              (4.72)

In finite dimensions, the dimensions of two dual vector spaces are equal.

dim  (𝒱) = dim (𝒱 ∗)                              (4 .73)

Definition 50 (Cotangent vector space) The dual space of the tangent vector space is called the cotangent vector space   ∗
T  MP  .

In the case of infinite-dimensional vector spaces such as the spaces of functions, a slight modification is required. When dealing with infinite dimensions, the topological dual space is defined by demanding the linear functionals to be continuous. In finite dimension this demand is not required explicitly, since all linear functionals are inherently continuous.

Before proceeding further it is necessary to clarify another piece of terminology connecting Definition 40 and Definition 16.

Definition 51 (Vector bundle) A fiber bundle (Definition 40), where the fibers F carry the structure of a vector space (Definition 16), is called a vector bundle.

Definition 52 ((Co)Tangent bundle) The (co)tangent spaces T MP  (  ∗
T  MP  ) (Definition 48) are parametrized on the points P ∈ 𝔐  . The disjoint union of all (co)tangent spaces

      TM  =      T MP                             (4 .74)
(                   )
   ∗      ⋃     ∗
  T M  =      T  MP                               (4 .75)
together with a projection
   πM  : T M →  𝔐                               (4 .76)
(π∗  : T ∗M → 𝔐  )                              (4 .77)
is a vector bundle (Definition 51), called the tangent bundle of 𝔐  . The manifold is the base space, while the tangent spaces at each of the points P are the attached fibers.

The structure of a (co)tangent bundle TM (  ∗
T  M ) of a smooth manifold 𝔐   (Definition 37) is again a smooth manifold.


Figure 4.4: Illustration of a tangent fiber and the tangent bundle of a manifold.

A particular mapping between the tangent and co-tangent bundle will be interesting in conjunction with Section 5.3 and is thus introduced here.

Definition 53 (Legendre map) The Legendre map is a mapping between the tangent and the cotangent bundle (Definition 52) of a manifold 𝔐   (Definition 37)

ˆL : T M →  T ∗M                               (4.78)
such that the projection map π1   (Definition 25)
π1 ∘L  = π1.                                (4.79)
Particularly, with         −1
v,w ∈  π  (x)  vectors from a fiber over x it is related to a function
L : T M →  T ∗M                               (4.80)
⟨ˆL(v),w ⟩ = dt|0L(v + tw ).                         (4.81)

Similar to the introduction of the elements of the dual vector space, the 1-forms, it is possible to define more complex mappings not only from the vector space but also its dual to the field above which the vector space has been constructed.

Definition 54 (Tensor) A multilinear mapping (linear every argument)

trs : 𝒱 × ...× 𝒱 × 𝒱∗ × ...× 𝒱 ∗→  ℝ                     (4.82)
    ◟---◝◜----◞   ◟----◝◜-----◞
         s              r
is called an (r,s)  -tensor.

In this context the original vectors and 1-forms appear as the special cases of (1,0)  and (0,1 )  tensors respectively. The collection of all tensors of type (r,s)  based on the vector space 𝒱 again has the structure of a vector space and shall be denoted by 𝒯sr(𝒱)  . The dimension of this vector space is

dim 𝒯s (𝒱 ) = r + s.                           (4.83)

While the bases of the tangent and co-tangent vector spaces can be linked to the charts of the manifold 𝔐 , this is so far not available for the vector space of tensors. It is therefore desirable to have a mapping available which, among other things, enables the determination of bases for the tensors spaces of arbitrary dimension from the initial tangent and co-tangent spaces.

Definition 55 (Tensor product) The non-commutative and associative mapping

      r       t         r+t
⊗ : 𝒯s (𝒱 ) × 𝒯u(𝒱 ) → 𝒯s+u (𝒱 )                      (4.84)
where the components follow
ξ∈𝒯r(𝒱),ζ ∈ 𝒯 t(𝒱) : (ξ ⊗ ζ)(v1,...,vs,α1,...,αr,w1, ...,wu,β1, ...,βt) =  (4.85a)
s      u
                   ξ(v1,...,vs,α1,...,αr)ζ(w1, ...,wu,β1, ...,βt)    (4.85b)

is called a tensor product.

4.6.1 Tensor Coordinates

Tensors with their algebraic structures represent a very abstract and powerful concept with considerable applicability.The notion of a tensor and its properties do not rely on the particular choice of its representation. The properties do not specify, however, how a particular tensor is defined and how to obtain its corresponding value for a given input. Coordinates with respect to a given choice of base b provide a means to address this problem.

The linear structure of tensors and using the tensor product (Definition 55) allows to express any tensor ξ in the form of

ξ ∈ 𝒯 r(𝒱 )                                    (4 .86)
ξ =    Ξ1.1.....rsb11......sr,                              (4 .87)
where Ξ1...r
1...s  are the coordinates of the tensor ξ in the base elements b1...s
 1...r  . Providing the full set of tensor coordinates along with the base they refer to, completely specifies the tensor. This important fact is quite commonly abused attributing special properties to the coordinates themselves and in fact considering tensors and vectors alike as little more then a suitable collection of values as found, for example, in matrix representations. While indispensable, a reduction to this primitive level of abstraction belies the true structure of the entities at hand. A tendency of this reductionism has even been compared to a mathematical disease [76]. Where this regression of abstraction may be tedious from a theoretical setting, its severity only increases in the case that properties should be qualified from the coordinates, when they are subjected to calculations of limited precision. It is therefore highly desirable to retain as much abstract information as possible in both theoretical as well as practical fields.

4.6.2 Bra-Ket Notation

As it is prominently used in quantum physics (see Section 5.7) a few words shall be invested to address the topic of Dirac’s bra-ket notation [77], which allows rapid coordinate manipulation of entities. Dirac identified a vector v of a vector space 𝒱  (Definition 16) with a so called ket, represented as:

|v⟩ ∈ 𝒱                                   (4.88)
At the same time a 1-form       ∗
α ∈ 𝒱  (Definition 49) is identified by a so called bra, written in the following form:
⟨α| ∈ 𝒱∗                                  (4.89)
The simplicity stems from the fact that the scalar product may simply be written as:
⟨α |v⟩,α ∈  𝒱 ,v ∈ 𝒱                             (4.90)
While this expression is highly similar to (4.70), it can further be enhanced as follows:
⟨α|A |v ⟩,α  ∈ 𝒱∗,v ∈ 𝒱                            (4.91)
Where A is a map of the form
A : 𝒱 →  𝒱                                 (4.92)
and in this context is often called an operator.

Furthermore it is possible to express the tensor product (Definition 55) by simply reversing the order of notation:

|v ⟩⟨α | = v ⊗ α, α ∈ 𝒱 ∗,v ∈ 𝒱                       (4.93)

The simplicity to describe these, in quantum physics regularly required, tasks in a concise, elegant and coordinate independent manner is the true expressive power made available by Dirac’s notation.

4.6.3 Beyond 1-Forms

The tensor product (Definition 55) allows the definition of an associative, non-commutative, graded algebra (Definition 18).

Definition 56 (Tensor algebra) The direct sum of all spaces 𝒯 r(𝒱 )

𝒯 (𝒱 ) =     𝒯 r(𝒱)                             (4.94)
together with the tensor product forms the tensor algebra.

Since there is no restriction on any of the indices r or s , the dimension of this tensor algebra is infinite.

While the tensor algebra is highly versatile and adaptable, its structure is not quite suited to conveniently represent physical quantities. A different structure is therefore introduced in the following. To this end, it is profitable to examine the symmetry properties of tensors. Since symmetry is determined by the exchange of the arguments of a tensor, only tensors which are purely drawn from either 𝒯0r (𝒱 )  or 𝒯0(𝒱)
s  are eligible for symmetry considerations.

Definition 57 (Symmetric Tensor) A tensor is called symmetric in case that an exchange of two of its arguments does not change its value. Considering both kinds of eligible tensor types gives

 ξ ∈ 𝒯p : ξ(...,v,..., w,...) = ξ(...,w,...,v,...),           (4.95a)
x ∈ 𝒯 p : x(...,α, ...,β,...) = x (...,β,...,α,...).          (4.95b)

The definition of anti-symmetric tensors follows analogously.

Definition 58 (Anti-Symmetric Tensor) A tensor is called anti-symmetric or skew symmetric in case that an exchange of two of its arguments reverses the sign of its value. Again considering both kinds of eligible tensor types gives

 ξ ∈ 𝒯 0: ξ (...,v, ...,w,...) = − ξ(...,w,...,v,...),          (4.96a)
x ∈ 𝒯 0 : x(...,α, ...,β,...) = − x (...,β,...,α,...).         (4.96b)

The cases which lack the degrees of freedom to accommodate the previous notion of symmetric and anti-symmetric are defined as being symmetric (Definition 57) as well as, at the same instant, anti-symmetric (Definition 58). In particular this concerns tensors of types 𝒯10(𝒱)  , 𝒯 10 (𝒱 )  and 𝒯00(𝒱)  .

The special case of anti-symmetric tensors of type   0
𝒯p (𝒱 )  is further distinguished by identifying them.

Definition 59 (p -form) A totally anti-symmetric tensor ξ ∈ 𝒯 0p  is called a p-form. The collection of all p-forms is denoted by ∧p 𝒱 ∗ .

∧p  ∗     0
   𝒱  ⊆ 𝒯p (𝒱)                               (4.97)

The anti-symmetry limits the dimensionality of the non-trivial subspaces and links them to the dimension n of the underlying vector space 𝒱  (Definition 16), since the anti-symmetry forces tensors to vanish in case

        ∧m   ∗               ∗
ξ = 0 ∈     𝒱 ,  ∀m  >  dim 𝒱                         (4.98)
thus resulting in
                         (  )
    ∧p  ∗   ----n!----    n
dim   𝒱   = (n − p)!p! =   p .                       (4.99)
Furthermore, the union of these subspaces
∧       ⋃ ∧
  𝒱 ∗ =     p𝒱 ∗                             (4.100)
is no longer closed under the tensor product (Definition 55). It is however possible to define a binary relation which is closed in the union of the fully anti-symmetric tensors.

Definition 60 (Exterior product) The exterior product

    ∧       ∧       ∧
∧ :  p𝒱 ∗ ×   q𝒱 ∗ →   p+q𝒱 ∗                       (4.101)
is a non-commutative, due to
             pq           ∧p   ∗     ∧q  ∗
α ∧ β = (− 1)  β ∧ α,  ∀ α   𝒱 ,β ∈    𝒱                  (4.102)
binary relation.

The space of fully anti-symmetric tensors (forms) ∧   ∗
  𝒱 together with the exterior product form an associative, non-commutative graded algebra (Definition 18).

The exterior product and the tensor product are related to one another according to

                    1  ∑
(α∧β)(v1,...,vp+q) =-----    sign(π)(α ⊗ β )(vπ(1),...,vπ(n+m ))        (4.103)
                  n!m!   π
                  ∀α ∧p 𝒱∗,β ∈ ∧q 𝒱 ∗,
where π are permutations of the input vectors and sign(π )  is the sign of a permutation.

4.6.4 Fields

After the basic geometric entities have been introduced in a per point nature it is desirable to also provide notions of distributed geometric items.

Definition 61 (Scalar field) A smooth mapping ω assigning to each point P of a differentiable manifold 𝔐  (Definition 37) a value of ℝ  is called a scalar field,

ω : 𝔐  → ℝ.                                (4.104)

Recalling that charts are bijections (Definition 24), it is possible to pull the scalar field ω back (Definition 44) to open sets of  n
ℝ  using

ω ∘ κ−1 : ℝn → ℝ.                             (4.105)
This provides an opportunity to define continuity of the scalar field ω by requiring continuity on ℝn  . Similarly, differentiability of the scalar field ω is defined by demanding the existence of the partial derivatives of the coordinate functions  i
x  .

The scalar fields on 𝔐  together with the binary operations of addition and multiplication form a ring 𝔽(𝔐)  (Definition 13) or 𝔽  for short over the manifold.

Using the scalar fields 𝔽 (𝔐 )  it is possible to define a derivation (Definition 27)

v : 𝔽(𝔐 ) → ℝ,                               (4.106)
which is isomorphic to the definition of vectors presented before (Definition 48) and is therefore suited as an alternate definition. This enables to view a vector as a directional derivative of a scalar field at a point P .
v (ω )(P),   P ∈ 𝔐,  ω ∈ 𝔽(𝔐 ),v ∈  TMP                   (4.107)

As scalar values are now associated to a manifold 𝔐  as well as a means of to derive vectors from the scalar fields, it is also desirable to extend this concept to vectors.

Definition 62 (Vector field) A vector field is defined as a mapping assigning to each point P  ∈ 𝔐  a vector from the tangent space T MP   (Definition 48) at P

V  : 𝔐 →  T MP .                             (4.108)

Thus any vector field is a section (Definition 41) of the tangent bundle (Definition 52).

Where a vector has been introduced as a mapping of the form 𝔽(𝔐  ) → ℝ  , a vector field is a mapping of the form 𝔽(𝔐 ) → 𝔽 (𝔐 )  . Thus a vector field appears as a derivation (Definition 27) of scalar field giving rise to the new scalar field ωn  :

ωn  = V (ω),  ωn, ω ∈ 𝔽(𝔐  )                       (4.109)
A vector field V , which indeed maps back to the scalar fields 𝔽(𝔐 )  , is said to be differentiable.


Figure 4.5: Integral curves γ (t)  of a vector field.

Having established the concepts of both differentiable curves (Definition 46) and vector fields (Definition 62) it is possible to link them.

Definition 63 (Integral curves (of vector fields)) A vector field V  (Definition 62) on a manifold 𝔐  (Definition 37) assigns a vector VP ∈ TP M to each point P  ∈ 𝔐  . Considering this vector VP  as a representative of the differential of a curve γ  (Definition 46) at P defines a set of integral curves or streamlines of the vector field V as illustrated in Figure 4.5.

˙γ  =  V ,   ∀P ∈ 𝔐                             (4.110)
 P     P

The existence of integral curves of a vector field allows to further qualify a vector field, since it links the extended entity of a curve to the local entity of a vector.

Definition 64 (Complete vector field) A vector field V  (Definition 62) is said to be complete, if the integral curves (Definition 63), which are initially only defined locally s∈I⊂ℝ , can be extended to all parameters s ∈ ℝ  for all points P ∈ 𝔐  .

Now that vector fields and curves have been intrinsically connected to each other, it is prudent to reexplore the issue of differing parametrization. Given two curves γ1  and γ2  with identical images in the manifold 𝔐  but differing parametrization

γ2(t) = γ1(σ(t))                             (4.111)
results in different values of the tangent vector
γ˙2 = σ˙˙γ1.                                (4.112)
This, however, indicates that the special cases of ˙σ = 1  such as reparametrizations of the form ′
t = t + τ do not change the obtained tangent vectors.

Similar to vector fields, it is also possible to associate a tensor to every point of a given manifold, thus leading to the next definition.

Definition 65 (Tensor field) A tensor field is defined as a mapping assigning to each point P ∈ 𝔐  a tensor from the tangent tensor algebras   r
𝒯s (𝒱)   (Definition 56) at P

V : 𝔐  → 𝒯 r(𝒱 ).                             (4.113)

Considering a vector field V on a manifold 𝔐  defines a system of integral curves, which completely fill the manifold without intersecting.

Definition 66 (Local flow) The integral curves of a vector field (Definition 63) give rise to a mapping

Φt : 𝔐  →  𝔐,                                (4.114)
which depends on one parameter t and describes a displacement of all points along the local integral curves, shown in Figure 4.5. This so called flow is considered local, when the mapping is defined for a limited range of the parameter t . It may also be written as
Φ  : 𝔐 × ℝ →  𝔐.                              (4.115)

Where the previous definition was only concerned with local properties, an extension to global scale is possible as well.

Definition 67 (Global flow) A local flow (Definition 66) resulting from a complete vector field V  (Definition 64) and therefore an unconstrained parameter t ∈ [− ∞; ∞ ]  is called a global flow or simply flow.

A global flow is a fibration (Definition 39) of the manifold 𝔐  on which it is defined along one-dimensional, non-intersecting sub manifolds, the integral curves (Definition 63) of the associated vector field (Definition 62).

Flows and vector fields correspond to each other bijectively (Definition 24): each vector field may be viewed as generating a field by its integral curves, while the corresponding vector field is recovered from a given flow by differentiation. Furthermore, flows can be combined

Φs+t = Φs ∘ Φt,                              (4.116)
which endows flows on 𝔐  with a group structure (Definition 11) where the identity element is
Φ0                                    (4.117)
a one parameter group.

Flows may also be combined with functions f on the manifold 𝔐  .

 P|-------Q                                 (4.118)
f |   f∗t=f∘Φ−t 1
f(P )

4.6.5 Derivatives

The structure of flows will prove to be of importance in further considerations in Section 5.3. It is therefore prudent to further explore relations of flows. As has already been established flows and vector fields (Definition 62) are linked by differentiation, whereby similar structures instantiating Definition 27 become apparent.

A flow Φt  (Definition 67) defines a pull back (Definition 44) of the tensor fields on the manifold 𝔐  (Definition 37)

  ∗   r          r
Φ t : 𝒯s (𝔐 ) → 𝒯s (𝔐 )                         (4.119)
called a Lie transport. A tensor field A ∈ T rs  , which is invariant under a flow
Φ∗A =  A                                 (4.120)
is called Lie dragged. This invariance, however, is not the general case, which motivates the next definition.

Definition 68 (Lie derivative) The measure of non-invariance of a tensor field with respect to the effect to the pull back due to a flow is obtained by the Lie derivative which is defined as

ℒv : 𝒯 r(𝔐 ) → 𝒯 r(𝔐 )                          (4.121)
      s         s
ℒv (A) = -d-|(Φ ∗(A )),                          (4.122)
         dt 0   t
where v is the vector field (Definition 62) generating the flow Φt   (Definition 67).

In the case of functions, which are represented as tensors of class 𝒯 0(𝒱 )
 0  , the Lie derivative yields

ℒv (f) = v(f),                               (4.123)
the directional derivative.

Since the special case of vector fields (Definition 62) is often encountered, it is awarded special notation, which shall not go omitted here, as it will also be used, especially in Chapter 5, which deals with dynamics.

Definition 69 (Lie bracket) The Lie derivative (Definition 68), in the case of vectors, which are included in Definition 68 as 𝒯01(𝒱 )  tensors, is used to define the Lie bracket, also known as the Jacobi bracket or commutator, as:

ℒv(w ) = [v, w ]                              (4.124)

The Lie bracket is skew-symmetric

[v, w ] = − [w, v]                              (4.125)
and satisfies the so called Jacobi identity
0 = [[v,w ],u] + [[u,v ],w ] + [[w, u ],v].                (4.126)
In case the Lie bracket vanishes,
[v, w ] = 0,                                (4.127)
the vector fields v and w are said to commute. Figure 4.6 gives a visualization of the commutator. Depending on the order in which the integral curves are followed, either the point D or the point C is reached, only in case the vector fields commute, the figure is closed at the point Q , otherwise, the Lie bracket yields by how much the closure falls short.


Figure 4.6: Illustration of the commutator [v, w ]  of two vector fields v and w .

Furthermore, there is a relation between the Lie bracket and the Lie derivative.

ℒ [v,w] = [ℒv, ℒw ] = ℒvℒw  − ℒw ℒv                     (4.128)

The connections of flows with vectors and the Lie derivative enables the definition of an exponential function exp using the following characteristics

γv (1 ) = exp (v),                            (4.129)
γ˙v (0 ) = v.                                  (4.130)
This leads to the following expression which also meets the group structure established in Equation 4.116,
Φt =  exp(tv),                               (4.131)
called the exponential map. Regarding the pull back (Definition 44) Φ∗t  the following expression can be established:
Φ ∗= exp (tℒv)                              (4.132)

Where the Lie derivative (Definition 68) is specially connected to flows and is a map among tensors (Definition 54) of the same type, mappings among different types of tensors and forms are also needed. They are presented in the following along with their connection to the Lie derivative.

A mapping especially important among p -forms (Definition 59) is provided in the following form of:

Definition 70 (Exterior derivative) A mapping

   ∧     ∧
d :  p →   p+1                              (4.133)
of p -forms to (p+ 1 )  -forms with the following properties

is called the exterior derivative. It is a (anti)derivation of degree 1.

While the exterior derivative has special importance to forms (Definition 59), a mapping only involving mixed tensors is also useful.

Definition 71 (Contraction) Given a tensor t ∈ T r
     s   (Definition 54) for s,r ≥ 1  the contraction k
Cl  is defined as a linear map

  k    r     r− 1
C l : Ts → Ts−1                              (4.134)
resulting in a tensor whose rank has been reduced by 2  .

As a particular special case the contraction can be used to express the scalar product given by Equation 4.70. Given a vector v and a 1-form α the scalar product of these two entities can be expressed using the tensor product (Definition 55) and the just introduced contraction as

C00(α ⊗ v) = ⟨α, v⟩.                           (4.135)
The tensor product first produces a tensor of type 𝒯 1(𝒱 )
 1  , whose contraction is of type 𝒯 0(𝒱 )
  0  which corresponds to scalar values.

Where this contraction is defined to act on a single tensor of mixed type, it is also possible to provide a similar and explicit mapping of forms involving an explicitly provided vector field (Definition 62).

Definition 72 (Interior product) Given a vector field x  (Definition 62) on a manifold 𝔐  (Definition 37) a linear mapping of the form

    ∧      ∧
ix :  k →   k− 1                             (4.136)
on the differential forms α ∈ ∧k  is called the interior product.

Written in component vectors it takes the shape:

(ixα )(v0,...,vk−1) = α (x, v0,...,vk− 1)                  (4.137)
For 1 -forms it takes the simple form:
i  = α (x )                                (4.138)

The interior product has similar properties to the exterior derivative (Definition 60) d  such as antisymmetry

ixiyα = − iyixα,                              (4.139)
ixix = 0,                                  (4.140)
and in conjunction with the exterior product (Definition 60) of a p -form α and a q -form β .
ix(α ∧ β) = (ix α) ∧ β + (− 1)kα ∧ (ixβ)                  (4.141)
For these reasons it is also often called an antiderivative of degree − 1  .

With the given properties the Lie derivative (Definition 68), the exterior derivative (Definition 70), and the interior product (Definition 72) can be linked in the expression

ℒx ω = d (ixω ) + ixd ω,                          (4.142)
which is known as Cartan’s identity or also homotopy (Definition 32) formula.

The exterior derivative allows to qualify differential forms (Definition 59), where the nomenclature is provided in the following definitions.

Definition 73 (Closed form) A form α  (Definition 59) with vanishing exterior derivative (Definition 70)

dα = 0                                  (4.143)
is called a closed form.

Definition 74 (Exact form) A form α  (Definition 59) is called exact, if it can be expressed by the external derivative (Definition 70) of another form β

α =  dβ                                  (4.144)

From the nilpotence of d  it follows that every exact form (Definition 74) is closed.

4.6.6 Special Spaces

Having established definitions for all components required for the treatment of geometrical problems, they are now assembled into the final settings used for the considerations, which are due to their importance specially named.

The further qualification of differentiable manifolds (Definition 37) is by pairing them with specific tensor fields (Definition 65).

Definition 75 (Metric tensor) A non-degenerate, symmetric bilinear form g ∈ 𝒯 0(𝒱 )
      2  is called a metric tensor.

Definition 76 (Metric tensor field) A tensor field (Definition 65) comprised entirely of metric tensors (Definition 75) is called a metric tensor field.

The availability of a metric tensor field allows to define the following:

Definition 77 (Inner product) A metric tensor field g on a differentiable manifold (Definition 37) defines a non-degenerate bilinear mapping

        (⋅,⋅) : TMP ×  TMP   → ℝ                          (4.145a)

(v, w ) =gP (v, w ),  ∀v,w  ∈ T MP ,∀P  ∈ 𝔐                (4.145b)

for the tangent space at every point. This mapping is called the inner product on the manifold 𝔐 .

Equipping a manifold with an inner product, defines the notion of orthogonality of two vectors in the tangential spaces at each of the points. This structure is of such significance, it warrants the following definition.

Definition 78 (Riemannian manifold) The pair (𝔐, g)  of a differential manifold 𝔐  (Definition 37) on which a positively definite metric tensor field g  (Definition 76) is defined and hence an inner product is available, is called a Riemanninan manifold.

Riemannian manifolds carry sufficient structure to define angles between vectors in each of the tangential spaces, but also allows for the definition of the concept of how far any two points of the manifold are apart, which is explored further in Section 4.8.

A Riemannian manifold, where all tangent spaces (Definition 48) are identical and which can therefore be covered by a single chart (Definition 36), is a very important special case encountered in everyday perception and has been the foundation of important developments in physics, as illustrated in Section 5.1.

In this case the structure can be simplified considerably, as can be experienced in the following two definitions.

Definition 79 (Affine space) A set of points 𝔄  accompanied by a vector space 𝒯  (Definition 16) is called an affine space, if the following assertions hold:

𝒯 is called the tangent space of 𝔄  .

The nomenclature in the case of the affine space is reminiscent of the manifold, however, the previously locally varying structures are now constant over all of the considered space. With these requirements an affine space inherently supports the notion of parallel lines. It however lacks structure to define lengths and angles.

Adding the structure of an inner product (Definition 77) remedies this deficiency and results in:

Definition 80 (Euclidean space) An affine space (Definition 79), where the tangent space 𝒯  (Definition 48) is equipped with an inner product which defines a norm, is called an Euclidean space ℰ .

Similarly to the symmetric metric structure (Definition 76) yielding the concepts of length and angles, a skew-symmetric structure has a distinct application as is illustrated in Section 5.3. The required structure begins by a simple definition.

Definition 81 (Symplectic form) A non-degenerate, closed (Definition 73), skew-symmetric bilinear form ω is called a symplectic form.

This definition is now used to qualify a manifold similarly to the Riemannian (Definition 78) case.

Definition 82 (Symplectic manifold) The pair (𝔐, ω )  of a differentiable manifold 𝔐  (Definition 37), where a symplectic form ω is defined in every tangent space T MP  , is called a symplectic manifold.

Due to the non-degeneracy together with the demand of skew-symmetry all symplectic manifolds necessarily have even dimension.

Definition 83 (Symplectic map) A diffeomorphism (Definition 38)

f : 𝔐  →  𝔑                                (4.149)
is called a symplectic map or symplectomorphism, if (𝔐,  ω)  and (𝔑, η)  are symplectic manifolds (Definition 82). The symplectic form can be expressed as a pullback (Definition 44).
f∗η = ω                                  (4.150)

Connected with symplectic spaces is an important theorem regarding the representation of the symplectic form. It is the basis for canonical representations in Section 5.3.

Definition 84 (Darboux’s theorem) Given a 2n -dimensional symplectic manifold (𝔐,ω)  (Definition 82) the symplectic form ω  (Definition 81) can always be rendered in the canonical form

ω =     dpi ∧ dqi                            (4.151)
in a neighbourhood U for every point x ∈  𝔐  .

Definition 85 (K¨ahler manifold) A complex manifold 𝔐  which has both a Riemannian and symplectic structure is called a K¨ahler manifold. It is required that the metric structure g and symplectic structure ω are compatible in the following manner:

g(ix,y ) = ω(x,y )                            (4.152)
Equivalently, the space may be viewed as a being equipped with a Hermitian inner product h composed as
h(x,y) = g (x, y) + ω(x,y ).                       (4.153)

4.6.7 Peculiarities in Special Spaces

The existence of a non-degenerate bilinear form, metric (Definition 76) or symplectic (Definition 81), allows to define an isomorphism between the tangent and co-tangent bundles of a manifold. This is accomplished by associating to every vector v tangent to 𝔐  at a point x ∈ 𝔐  one form by

gv(w ) = g(v,w )                             (4.154)
in the case of a Riemannian metric and
ωv(w ) = ω(v, w)                             (4.155)
in the symplectic case. Thus a vector v defines a 1  -form and is thereby linked to it, defining an isomorphism.
I : T M  →  TM                               (4.156)
The non-degenerate nature of the used bilinear form ensures that this relation also uniquely associates a vector with a 1  -form. The existence of such a mapping is essential as it also provides a structure with which to translate different kinds of tensors into one another.

Furthermore it is possible to identify flows and their associated vector fields which preserve the metric or symplectic structures. Using the Lie derivative this reads

ℒ  g = 0                                 (4.157)
in case of the a metric structure g . The vector fields meeting this requirement are called Killing vector fields [74 ]. In the symplectic case the expression is almost identical with the symplectic form taking the place of the metric
ℒv ω =  0.                                (4.158)
Complying vector fields are called Hamiltonian in this case. It is useful to further examine the symplectic case by expanding the expression using the homotopy formula
ℒv ω = d (ivω ) + ivd ω = 0.                       (4.159)
Since ω is closed and hence dω =  0  , it follows
d(ivω) = 0                                (4.160)
thus indicating
ivfω = − df,                                (4.161)
where f is a function on the symplecetic manifold 𝔐  . The function f generates the vector field vf  . The definition of vF  in Equation 4.161 is implicit and may be converted to an explicit form using the isomorphism defined in Equation 4.156 to obtain
vf =  Idf.                                (4.162)
This explicit form can be combined with the Lie bracket (Definition 69) to obtain additional structure.

Definition 86 (Poisson bracket) The Poisson bracket is obtained by applying the Lie bracket for vector fields to an expression due to the established isomorphism (Equation  4.156).

[Idf,Idg ] = [vf ,vg] = v{f,g}                       (4.163)
The Poisson bracket or commutator of functions on a manifold 𝔐  is a bilinear, skew symmetric map satisfying the Jacobi identity.

A manifold 𝔐  along with a Poisson structure is called a Poisson manifold. Every symplectic manifold is also a Poisson manifold, while the reverse is not necessarily true, as the Poisson structure may be degenerate, thus being more general. In the symplectic case, the Poisson structure and the symplectic structure are inverse to each other.

P ∘ ω = − id                               (4.164)