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 to a diﬀerentiable manifold (Deﬁnition 37) is called a curve:

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

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

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 is considered to be diﬀerentiable, if the functions describing its coordinate representations 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.

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

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 , the identity of the ﬁrst derivatives deﬁnes a set of equivalence classes (Deﬁnition 9):

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 , its dual space is comprised by the linear functionals

The dual space 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.

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

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

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 carry the structure of a vector space (Deﬁnition 16), is called a vector bundle.

Deﬁnition 52 ((Co)Tangent bundle) The (co)tangent spaces () (Deﬁnition 48) are parametrized on the points . The disjoint union of all (co)tangent spaces

The structure of a (co)tangent bundle () of a smooth manifold (Deﬁnition 37) is again a smooth 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 (Deﬁnition 37)

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.

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

While the bases of the tangent and co-tangent vector spaces can be linked to the charts of the manifold , 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

is called a tensor product.

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 provide a means to address this problem.

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

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 of a vector space (Deﬁnition 16) with a so called ket, represented as:

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

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.

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

Since there is no restriction on any of the indices or , 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 or 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

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

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 , and .

The special case of anti-symmetric tensors of type is further distinguished by identifying them.

Deﬁnition 59 (-form) A totally anti-symmetric tensor is called a p-form. The collection of all p-forms is denoted by .

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

The space of fully anti-symmetric tensors (forms) 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

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 assigning to each point of a diﬀerentiable manifold (Deﬁnition 37) a value of is called a scalar ﬁeld,

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

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

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

As scalar values are now associated to a manifold 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 a vector from the tangent space (Deﬁnition 48) at

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 , a vector ﬁeld is a mapping of the form . Thus a vector ﬁeld appears as a derivation (Deﬁnition 27) of scalar ﬁeld giving rise to the new scalar 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 (Deﬁnition 62) on a manifold (Deﬁnition 37) assigns a vector to each point . Considering this vector as a representative of the diﬀerential of a curve (Deﬁnition 46) at deﬁnes a set of integral curves or streamlines of the vector ﬁeld as illustrated in Figure 4.5.

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 (Deﬁnition 62) is said to be complete, if the integral curves (Deﬁnition 63), which are initially only deﬁned locally , can be extended to all parameters for all points .

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 and with identical images in the manifold but diﬀering parametrization

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 a tensor from the tangent tensor algebras (Deﬁnition 56) at

Considering a vector ﬁeld on a manifold 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

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 (Deﬁnition 64) and therefore an unconstrained parameter is called a global ﬂow or simply ﬂow.

A global ﬂow is a ﬁbration (Deﬁnition 39) of the manifold 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

Flows may also be combined with functions on the manifold .

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

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

In the case of functions, which are represented as tensors of class , the Lie derivative yields

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 tensors, is used to deﬁne the Lie bracket, also known as the Jacobi bracket or commutator, as:

The Lie bracket is skew-symmetric

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

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

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 -forms (Deﬁnition 59) is provided in the following form of:

Deﬁnition 70 (Exterior derivative) A mapping

- is linear.
- is the diﬀerential for smooth functions .
- – is nilpotent.
- – obeys a graded Leibniz rule.

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 (Deﬁnition 54) for the contraction is deﬁned as a linear map

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

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 (Deﬁnition 62) on a manifold (Deﬁnition 37) a linear mapping of the form

Written in component vectors it takes the shape:

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

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

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 (Deﬁnition 59) with vanishing exterior derivative (Deﬁnition 70)

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

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

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 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 on a diﬀerentiable manifold (Deﬁnition 37) deﬁnes a non-degenerate bilinear mapping

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

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 of a diﬀerential manifold (Deﬁnition 37) on which a positively deﬁnite metric tensor ﬁeld (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 accompanied by a vector space (Deﬁnition 16) is called an aﬃne space, if the following assertions hold:

- A mapping assigning to every pair of points to an element exists.
- For every pair of a point and vector there is exactly one point such
that
- For every three points it holds

is called the tangent space of .

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 (Deﬁnition 48) is equipped with an inner product which deﬁnes a norm, is called an Euclidean space .

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 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 of a diﬀerentiable manifold (Deﬁnition 37), where a symplectic form is deﬁned in every tangent space , 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)

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 -dimensional symplectic manifold (Deﬁnition 82) the symplectic form (Deﬁnition 81) can always be rendered in the canonical form

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

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 tangent to at a point one form by

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

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).

A manifold 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.