Elements of differential geometry
Let me start with a disclaimer regarding what this text is, and what it is not. This is by no means meant as an introduction to differential geometry that would be self-contained or com-prehensive in any sense. That would require writing a whole book, which I have no competence to do. Moreover, there are excellent textbooks on the subject on the market. Whoever reads this text should keep in mind that it is a collection of miscellaneous notes written by a physics practitioner for private use. As such, this is largely intended to provide a compilation of useful formulas to aid the memory, and a brief outline of the motivation behind the various concepts used in differential geometry. The focus is on intuition rather than rigor, and the presentation is accordingly biased in line with my personal taste. As a final word of caution, let me stress that in spite of the title, the text is not primarily aboutgeometry. The choice of topics included is guided by the aim to develop the basic methods of calculus on differentiable manifolds.
Tangent space, vector fields and their flow
I will take for granted the notion of a smooth function f on a manifold M, f : M →R. The next step is to introduce the concept of a derivative of the function. Should M actually be a Euclidean space, Rn, there is a standard solution, based on the naturally defined Cartesian coordinates: there are altogether n different partial derivatives of f with respect to the indi-vidual coordinates onRn,∂f /∂xµ,µ= 1, . . . , n. On a general manifold, such a preferred set of global coordinates does not exist, and thus a more geometric definition is required. The trick is to think of adirectional derivative instead of a set of partial derivatives with respect to fixed coordinates. On Rn, the object vµ∂f /∂xµ represents a directional derivative of f along the vector with coordinates vµ. On a curved manifold, where we do not have an a priori defined concept of a vector, we reverse the argument and take the expression vµ∂f /∂xµ itself as the new definition of a vector.
To be more precise, a tangent vectorat pointx∈ M is identified with a differential operator acting on test functions, defined in some neighborhood of x,
v =vµ ∂
∂xµ : f 7→v[f]≡v µ ∂f
∂xµ
x
. (1)
I used a specific local coordinate system to spell out the definition explicitly, but the concept clearly has a geometric meaning independent of the concrete coordinate system chosen. All tangent vectors to Mat a given point xspan a vector space, called the tangent space of M
at xand denoted as TxM. Obviously, the vectors ∂µ≡
∂
∂xµ, µ= 1, . . . ,dimM, (2)
define a basis of TxM, called the coordinate basis. It is, however, often convenient to have a definite basis of vectors, independent of the choice of local coordinates. Such anon-coordinate basiswill be denoted as ea ≡eµa∂µ, wherea= 1, . . . ,dimM. While coordinate bases are very useful in explicit calculations, it is the ea bases that have a coordinate-independent geometric meaning, and are therefore conceptually preferable.
misleading, and is based on the fact that whenever we can define a tangent vector more geomet-rically (that is, when M is a Euclidean space or a surface embedded in a higher-dimensional Euclidean space), there is a natural isomorphism between the spaces of such “geometric” tan-gent vectors and directional derivatives. On manifolds where we do not enjoy such a luxury, we only have the directional derivative (1). Whether we decide to call it a tangent vector is just a matter of taste or convention. Remember that, after all, we are trying to develop calculus on manifolds, not their geometry.
Following closely the physics terminology, a vector field is a smooth mapping which to each point x ∈ M assigns a tangent vector v(x) ∈ TxM. The collection of all vector fields on M
will be denoted asTM. A vector field on a manifold can be visualized in a way that avoids the necessity to draw “tangent vectors” to the manifold, namely by considering theflow generated by the field. This consists of a family of curves that we can think of as trajectories of motion of a fluid on M with v(x) being the velocity of the fluid in the Euler picture. Formally, this amounts to solving the set of differential equations
dγµ(t) dt =v
µ(γ(t)), (3)
where γ : R → M is the parameterization of a given trajectory. In some neighborhood of a chosen point on M, this set of differential equations has a unique solution, leading to a family of non-intersecting flow lines: the flow defines a diffeomorphism on M. Note that this picture also provides an alternative — and even more geometric than Eq. (1) — way to introduce a tangent vector at a single point x∈ M: it can be defined in terms of the curve passing through xthat the vector is tangent to. Since only the velocity of the fluid atxmatters for the definition of the tangent vector atx, the tangent vector formally corresponds to a whole equivalence class of smooth curves passing through this point.
Cotangent space, tensors, differential forms
Once we have successfully formulated the notion of tangent vectors and a tangent space, we can immediately build up a lot of additional structure. The cotangent space Tx∗M is defined as the dual of the tangent spaceTxM, its elements —cotangent vectorsor1-forms— are thus linear maps from TxMto R. A special class of 1-forms that have a simple geometric meaning is that of differentials of ordinary (scalar) functions. Formally, one defines, for f :M → R,
df(v)≡v[f] =vµ ∂f
∂xµ for v ∈TxM. (4) The differential df acts naturally as a cotangent vector by associating with a given vector the increment of the function f in its direction. A special case of such a differential of a function is the dual coordinate basis,
dxµ∈Tx∗M, dxµ(∂ν) =δµν. (5) Note that this offers a dual interpretation of the vector component vµ as vµ=v[xµ] = dxµ(v). A dual non-coordinate basis,e∗a ≡eaµdxµ, is defined analogously by its action on the vectorsea of the non-coordinate basis, e∗a(eb) = δba. A general 1-form ω ∈ T
∗
xM is then simply a linear combination of the basis elements, for instance in the dual coordinate basis
Having at hand both the tangent and the cotangent space, an arbitrary tensor at a given point x ∈ M, and a tensor field of an arbitrary type, can be constructed in a way well-known from linear algebra. Here I will just mention the important special case of fully antisymmetric tensor fields of type (0, p), also called p-forms. The components of a p-form ω in a coordinate or non-coordinate basis are defined by
ω = 1
p!ωµ1···µpdx
µ1 ∧ · · · ∧dxµp, ω= 1
p!ωa1···ape
∗a1 ∧ · · · ∧e∗ap, (7)
whereωµ1···µp and ωa1···ap are fully antisymmetric in all indices and the wedge∧denotes a fully
antisymmetrized tensor product, implying a sum over all permutations π of the indices, e∗a1 ∧ · · · ∧e∗ap =X
π
sgnπe∗aπ(1)⊗ · · · ⊗e∗aπ(p). (8)
This ensures that the component of the p-form can be obtained by the action of the form on a collection of basis vectors as expected; the prefactor 1/p! in Eq. (7) cancels against the sum over permutations in the wedge product,
ωa1···ap =ω(ea1, . . . ,eap). (9)
An exterior product is a linear, distributive and associative binary operation that maps a p-form ωp and a q-form ωq on a (p+q)-form ωp∧ωq. The result can be defined in an obvious manner in a specific basis using the expansion (7). The exterior product can, however, also be defined in a strictly geometrical way, independent of the choice of basis, through its action on a set of vectors,
(ωp∧ωq)(v1, . . . ,vp+q) = 1 p!q!
X
π
sgnπ ωp(vπ(1), . . . ,vπ(p))ωq(vπ(p+1), . . . ,vπ(p+q)). (10)
This makes it clear that the exterior product is graded-commutative, that is,
ωp∧ωq= (−1)pqωq∧ωp. (11) A natural extension of the definition of the exterior product to linear combinations of forms of different degrees gives the direct sum of spaces of p-forms with p= 0, . . . ,dimMthe structure of an associative algebra, the Grassmann algebra.
While the exterior product is a binary operation that increases the degree of the differential forms, there is a unary operation that acts in the opposite direction. The interior product
of a (p+ 1)-formω with the vector v is a p-form ιvω, defined by its action on a set of vectors,
ιvω(v1, . . . ,vp) = ω(v,v1, . . . ,vp). (12) This definition is geometric, that is independent of the choice of basis, but a similarly simple expression for the components of the interior product can be given,
(ιvω)a1···ap =v
bω
ba1···ap. (13)
The interior product satisfies the anticommutation propertyιuιv =−ιvιu, which impliesι2v = 0
for any vector field v. Moreover, ιv is an antiderivation: for any p-form ωp and any q-form ωq, one has
Maps between manifolds, push-forward and pull-back
Consider a smooth (possibly only locally defined) mappingφbetween two manifoldsMandN, φ:M → N. This naturally induces a mapping between the respective tangent and cotangent spaces. As to the former, simply think of mapping a tangent vector to a curve on M to a tangent vector to the image curve on N. Formally, one starts by associating with any test function g : N → R the function g ◦φ : M → R. For any v ∈ TxM, we then introduce its
push-forward as the element φ∗v ∈Tφ(x)N for which
φ∗v[g]≡v[g◦φ]. (15) It is important to stress that this is not a mere coordinate redefinition: the manifolds M and
N can be completely different and the map φ need not even be invertible. For such a general φ, the push-forward however does not promote to a mapping from TM to TN: the naively expected vector field on N is in general well-defined only at points on N that have a unique preimage under φ. This is guaranteed when φ is a diffeomorphism.
Similarly, for any ω ∈Tφ(x)∗ N, we define itspull-back as such an elementφ∗ω∈Tx∗Mthat φ∗ω(v)≡ω(φ∗v) for any v ∈TxM. (16) Unlike for the push-forward, there is no issue with extending the pull-back to a map from T∗N
to T∗M; this is thanks to the fact that every point x ∈ M has a unique image in N under the action of φ. A typical application of the pull-back concept in physics is with Mbeing the spacetime manifold and N being the manifold from which a fieldφ takes values, for instance a unit sphere in case of the nonlinear sigma model. Denoting the coordinates onM asxµ and a set of local coordinates on N asξm, we obtain from the definitions (15) and (16)
φ∗dξm(v) = φ∗v[ξm] =v[ξm◦φ]≡v[φm] =vµ∂µφm, or φ∗dξm =∂µφmdxµ. (17) The same result can be obtained naturally by naively “substituting the variables” in dφm. Note that the push-forward and the pull-back act in the opposite directions. One can therefore trivially generalize the push-forward to an arbitrary tensor of type (p,0), and the pull-back to an arbitrary tensor of type (0, p). There is, however, no natural definition of such a lift by the map φ : M → N for mixed-type tensors, unless the map φ is invertible, in which case a pull-back by φ−1 can be used to push-forward covariant tensors, and vice versa.
Lie transport and Lie derivative
A special class of maps that map a given manifold into itself is generated by the flow of vector fields. Denote by φv,t the map from MtoMthat solves the flow equation (3) with the initial condition that φv,0 is the identity map. In other words, if γ(0) =x ∈ M, then γ(t) =φv,t(x). Since the flow, as represented by the map φv,t, defines a diffeomorphism of M to itself at any fixed t, it can be used to “transport” tensor fields of any type around the manifold by a push-forward. This procedure is referred to as theLie transportalong the vector fieldv. The question how a given tensor field T varies under an infinitesimal Lie transport gives rise to the concept of a Lie derivative. This is formally defined by
£vT ≡ −
d(φv,t∗T) dt
t=0 = lim
ε→0
T −φv,ε∗T
The minus sign stems from the fact that the value ofφv,t∗T atx∈ Mcorresponds toT(φv,−tx), pushed forward to xvia Eq. (15). Let us see how this works in specific cases. For functions on
M we have simply φv,t∗f(x) = f(φv,−tx). Treating v as a differential operator that translates the argument of f, we readily recognize that the flow equation for f is formally solved by
φv,t∗f =e−vtf =⇒ £vf =v[f] =vµ∂µf. (19)
For vector fields, we have analogously φv,t∗u|x[f] = u|φv,−tx[f ◦φv,t]. This amounts to trans-lating f from x to φv,−tx, acting on it with the differential operator u, and translating the result back tox. In other words, the formal solution to the flow equation for a vector fieldu is φv,t∗u =e−vtuevt =⇒ £vu = [v,u], (20)
where I introduced the Lie bracketof two vector fields as their formal commutator as differ-ential operators. The same result can be obtained in a more conservative manner by focusing on an infinitesimal push-forward of the vector field: according to Eq. (15), we have
φv,ε∗u|x[f] = u|φv,−εx[f ◦φv,ε]
=
uµ(x)−εvν(x)∂νuµ(x) +O(ε2)
∂µf(x) +ε∂νf(x)∂µvν(x) +O(ε2)
=uµ(x) +ε[uν(x)∂νvµ(x)−vν(x)∂νuµ(x)] +O(ε2) ∂µf(x).
(21)
The Lie bracket of vector fields gives rise to a Lie algebra structure on TM. The associated Jacobi identity can be equivalently expressed as the Leibniz rule for the Lie derivative,
£u[v,w] = [£uv,w] + [v,£uw]. (22)
Once we know the Lie derivative of a function and a vector field, we can obtain the Lie derivative of any tensor field by imposing the Leibniz rule. To summarize the results for the practically most useful cases of a scalar field f, vector field u and 1-form ω, their Lie derivative along the vector field v is given by the following explicit expressions in a coordinate basis,
£vf =vµ∂µf =v[f],
(£vu)µ =vν∂νuµ−uν∂νvµ= [v,u]µ, (£vω)µ =vν∂νωµ+ων∂µvν.
(23)
These generalize for an arbitrary tensor fieldT as (£vT)αβ···µν··· =vλ∂λTµν···αβ···+T
αβ···
λν···∂µvλ+T αβ···
µλ···∂νvλ+· · · −Tµν···λβ···∂λvα−Tµν···αλ···∂λvβ− · · · . (24) On differential forms, one can in addition give an elegant coordinate-free expression for the Lie derivative. Specifically for a p-form, acting on a set of vectors vi, i= 1, . . . , p, one has
(£uω)(v1, . . . ,vp) =u[ω(v1, . . . ,vp)]− p
X
i=1
ω(v1, . . . ,[u,vi], . . . ,vp). (25) A special but nontrivial case of this formula describes the action of the vector fieldumultiplied by a scalar function f,
£fuω =f£uω+ df ∧ιuω. (26)
Finally, for the sake of completeness, I will mention a useful operator identity, expressing the interplay of the Lie derivative and the interior product on differential forms,
[£u, ιv] = [ιu,£v] =ι[u,v]. (27) Note that written as£u(ιvω) =ι[u,v]ω+ιv(£uω), this is just a special case of the Leibniz rule
Exterior derivative, calculus of differential forms
The Lie derivative has a simple geometric interpretation in terms of infinitesimal Lie transport as outlined above, and plays an important role for implementation of symmetries on manifolds. In spite of its geometric significance, however, it is not easy to use in practice as one mostly has to resort to coordinate-basis expressions as in Eq. (24). In addition, the Lie derivative £v
is, in fact, not a derivative in that it is not linear in the vector field v, as Eq. (26) shows. I will now introduce another notion of a derivative that operates on the Grassmann (exterior) algebra of differential forms, and is thus called the exterior derivative. The price to pay is the lack of intuition. Indeed, it is not immediately obvious why such a notion should exist at all, and what its geometric meaning is. This is, however, more than outweighed by the utility of this concept, which lies at the heart of both differential and integral calculus on manifolds. The exterior derivative is most easily defined in the coordinate basis,
ω = 1
p!ωµ1···µpdx
µ1 ∧ · · · ∧dxµp =⇒ dω≡ 1
p!(∂νωµ1···µp)dx
ν ∧
dxµ1 ∧ · · · ∧dxµp. (28)
This definition is useful for practical calculations, but does not make it manifest that dω is well-defined, that is, independent of the choice of the local coordinates. To prove this requires an inspection of the behavior of dω under a coordinate redefinition, an approach that is very common (to the point of obsession) in physics, but not developed in detail here. An explicitly coordinate-free definition of the exterior derivative can be given in terms of the so-called Cartan formula, by the action of dω on a set of vector fields,
dω(v1, . . . ,vp+1) = p+1
X
i=1
(−1)i+1vi[ω(v1, . . . ,vˆi, . . . ,vp+1)]
+X
i<j
(−1)i+jω([vi,vj],v1, . . . ,vˆi, . . . ,vˆj, . . . ,vp+1),
(29)
where the hat indicates arguments of ω that are omitted. For a 0-form (scalar function) f, a 1-form ω and a 2-form Ω, this formula reduces to the following practically useful expressions,
df(v) = v[f],
dω(u,v) = u[ω(v)]−v[ω(u)]−ω([u,v]), (30)
dΩ(u,v,w) = u[Ω(v,w)] +v[Ω(w,u)] +w[Ω(u,v)]−Ω([u,v],w)−Ω([v,w],u)−Ω([w,u],v). Equations (28) and (29) give two constructive definitions of the exterior derivative. They are by no means accidental. In fact, it can be shown that the exterior derivative is uniquely defined by requiring that: (i) it reduces to the ordinary differential on 0-forms, (ii) it satisfies ddω= 0 for any differential form ω, and (iii) it is an antiderivation just like the interior product, that is for any p-form ωp and any q-form ωq,
d(ωp∧ωq) = (dωp)∧ωq+ (−1)pωp∧(dωq). (31) The exterior derivative is natural with respect to pull-back, and hence also with respect to Lie derivative. Thus, for any smooth map φ, vector field v and differential form ω,
d(φ∗ω) = φ∗(dω), d(£vω) = £v(dω). (32)
The three natural derivations acting on differential forms, ιv, £v and d, are connected by the
“Cartan magic formula,”
Affine connection and covariant derivative
The concepts of Lie derivative and exterior derivative only require the differentiable structure that a manifold M is by definition endowed with. Here I shall introduce a different type of derivative, which will naturally lead to a new structure: the affine connection. The argument is based on the notion of avielbeinorframe field, that is a set of pointwise linearly independent but otherwise arbitrarily chosen vector fields ea(x) that span a non-coordinate basis of TxM at any point x∈ M. We will likewise need the dual vielbein e∗a(x) satisfying e∗a(eb) = δab. Let us for simplicity start with vector fields. To see how to define a derivative of a vector field, I will draw on the analogy with vector calculus in Euclidean space. There, it is most convenient to express the vector in a Cartesian basis: since the basis vectors do not vary throughout the space, taking a derivative of the vector field amounts to taking a derivative of its Cartesian components. I will therefore at firstassume that also on the given manifoldM, there is a basis
˜
ea that can be considered “constant” in a certain sense. The variation of an arbitrary vector field v(x) is then naturally defined through the variation of its components in this basis,
v = ˜vaea˜ =⇒ ∇µv = (∂µ˜va) ˜ea. (34) I use the symbols ∂µand ∇µ to denote an ordinary derivative and the desired covariant deriva-tive, respectively. The indexµis just a placeholder and does not necessarily refer to a coordinate basis; one can analogously define directional derivatives along any vector field. Since the com-ponents of a vector field in the vielbein basis are well-defined scalar fields on the manifold,∇µv in Eq. (34) is a manifestly well-defined tensor field.
Eventually, we would however like to express the components of the derivative of the vector v in an arbitrary vielbein basisea that is not necessarily constant. In other words, we are looking for (∇µv)a such that∇µv = (∇µv)aea. The new basis is related to the “constant” basis ˜ea by a local GL(n) transformation U, where n≡dimM,
ea= ˜eb(U−1)ba, va≡e∗a(v) = Uab˜vb. (35) Upon inserting this into Eq. (34), we obtain ∇µv=∂µ[(U−1)abvb]ecUca, and from there
(∇µv)a=∂µva+ (U ∂µU−1)abv b ≡
∂µva+ (Aµ)abv b
. (36)
Upon a further change of basis by a matrix g ∈ GL(n), U 7→ gU so that the matrix field Aµ changes togAµg−1+g∂µg−1. This closely parallels what is well known in gauge field theory. The role of the matrix Aµ is to ensure that under an arbitrary local change of the vielbein basis by g ∈GL(n), the component (∇µv)atransforms togab(∇µv)b, thus ensuring that the combination
∇µv = (∇µv)aeais independent of the choice of basis. In other words, the description of vectors in terms of the vielbein possesses a gauge redundancy with respect to GL(n) transformations of the vielbein, Aµ being the corresponding gauge field.
Motivated by the above observation, I will now give a generaldefinition of a covariant derivative of a vector field. In accord with the conventions used in gauge field theory, it is customary to introduce theaffine connectionas a matrix-valued 1-form field Ω≡Aµdxµ that is subject to the local action ofGL(n) via
The directional covariant derivative of the vector field v(x) along the vector field u(x) can then be expressed formally as
(∇uv)a=u[va] + Ωab(u)v
b, or (∇
µv)a=∂µva+ (Aµ)abv
b. (38) Covariant derivatives of other tensors are deduced by imposing the Leibniz rule. For instance, for a 1-form ω we find that the vielbein component of the covariant derivative reads
(∇uω)a =u[ωa]−ωbΩba(u), or (∇µω)a =∂µωa−ωb(Aµ)ba. (39) It is easy to convert the expressions (38) and (39) to a given coordinate basis. It is customary to express the result as
(∇µv)ν =∂µvν + Γνµλv
λ, (∇
µω)ν =∂µων −Γλµνωλ, (40) where Γλ
µν are theChristoffel coefficients, determined by the vielbein and the gauge connec-tion Aµ,
Γλµν =eλa[∂µeaν + (Aµ)abe b ν]≡e
λ
a∇µeaν, or ∇µ∂ν = Γλµν∂λ, (41) with a slight abuse of notation. The generalization of all the above expressions for the covariant derivative to tensors of arbitrary type is obvious,
(∇µT)ab···kl··· =∂µTkl···ab···+ (Aµ)aiT ib···
kl··· + (Aµ)biT ai···
kl··· +· · · −T ab···
il··· (Aµ)ik−T ab···
ki···(Aµ)il− · · · , (∇µT)
αβ···
κλ··· =∂µT αβ··· κλ··· + Γ
α µσT
σβ··· κλ··· + Γ
β µσT
ασ···
κλ··· +· · · −ΓσµκT αβ··· σλ··· −Γ
σ µλT
αβ···
κσ··· − · · · .
(42)
Note how the attempt to define a derivative of tensor fields naturally led to the introduction of a new structure on the manifold: the affine connection. The connection is not only a practical tool that allows one to construct manifestly geometric objects. It also provides a new insight into the intrinsic geometry of the manifold. The case of the flat Euclidean space, which I used to motivate the general definition of the covariant derivative, corresponds to a “pure gauge” connection, such that a vielbein basis exists in whichAµ = 0. Following the analogy with gauge field theory, it is then natural to characterize the general case with a nontrivial connection by the corresponding gauge-invariant “field strength,” that is the curvature 2-form,
Rab ≡dΩab+ Ωac∧Ωcb. (43) The coordinate-space components of this 2-form are then likewise given by the commutator of covariant derivatives in the vector representation ofGL(n), (Rµν)ab = [∇µ,∇ν]ab. In differential geometry, it is however more common to define the curvature tensor in terms of a commutator ofdirectional covariant derivatives. It is straightforward to show that for arbitrary vector fields u, v and w,
∇u∇vw− ∇v∇uw− ∇[u,v]w
a
=Rab(u,v)wb. (44) Thinking of the left-hand side as the differential operator [∇u,∇v]− ∇[u,v]acting on the vector w, the very fact that the curvature tensor can be nontrivial underlines the richer structure of the covariant derivative as compared to the Lie derivative, for which the corresponding operator, [£u,£v]−£[u,v], identically vanishes.
Another quantity that characterizes the intrinsic geometry of the manifold is the torsion 2-form. This is defined as the covariant exterior derivative of the dual vielbein,
Ta≡de∗a+ Ωab∧e∗b, Ta(u,v) = ∇uv− ∇vu−[u,v]
a
The latter expression, which is straightforward to prove, makes it clear that in the coordi-nate basis, the corresponding torsion tensor is just the antisymmetric part of the Christoffel coefficients. The curvature and torsion 2-forms, or tensors, cannot take arbitrary values, but satisfy a set of constraints known as the Bianchi identities. These are obtained by taking the covariant exterior derivatives of the two 2-forms, and read
dRab+ Ωac∧Rcb−Rac∧Ωcb = 0,
dTa+ Ωab∧Tb =Rab∧e∗b. (46)
Metric, Riemannian geometry
In order for a manifold Mto be a true generalization of a flat Euclidean space, it needs to be equipped with a notion of distance. Naively, one might want to define distance of two points as the length of a shortest path connecting them, but this may be tricky to implement if the topology of the manifold is nontrivial. It is instead common to work locally, and introduce the notion of a metric through the norm of tangent vectors. This is in turn most conveniently derived from a concept of inner product of tangent vectors. Formally, themetricis thus defined as a symmetric tensor field of type (0,2),
g=gµνdxµ⊗dxν =gabe∗a⊗e∗b, where gab =g(ea,eb), (47) and the resulting inner product of vectors u and v is then simply hu,vi ≡g(u,v) = gabuavb. The concepts of metric and affine connection have different geometric purposes, and so can in principle be introduced completely independently. However, once we do have the metric, there turns out to be a unique connection that is free of torsion and makes the metric covariantly constant,∇µg= 0, the so-calledLevi-Civita connection. In a coordinate basis, the condition of vanishing torsion translates into the symmetry of the Christoffel coefficients, Γλ
µν = Γλνµ. The coefficients can be expressed explicitly in terms of the metric as
Γλµν = 1 2g
λα
(∂µgαν +∂νgµα−∂αgµν). (48) Due to vanishing torsion, the second of the Bianchi identities in Eq. (46) reduces toRab∧e∗b = 0, or equivalently in terms of its action on vector fields u, v, w,
Rab(u,v)wb+Rab(v,w)ub +Rab(w,u)vb = 0. (49)
As is well known from linear algebra, having the concept of inner product at hand allows one to define a natural isomorphism between different types of tensors. Physicists usually speak of “raising” and “lowering” indices in this context. One can however do away with explicit index expressions by making use of the musical isomorphisms between TM and T∗M. Thus, to every vector field, v ∈ TM, one can associate its flat v[ ∈ T∗M, defined by its action on an arbitrary vector u,
v[(u)≡ hv,ui. (50)
This is a formalization of the operation of lowering an index. The inverse operation associates with a 1-form ω∈T∗Mits sharp ω]∈TM, defined by
This is a formalization of the operation of raising an index. The definition of musical isomor-phisms can be naturally extended to tensors of other types.
Another type of isomorphism between spaces of tensors of different types, known from vector calculus, is constructed using the fully antisymmetric Levi-Civita symbol, µ1µ2µ3···, defined so
that123···= 1. This type of isomorphism is generalized to all differential forms on an arbitrary
orientable (see the section on integration below for a definition) manifold by the Hodge star dual. This maps ap-form on an (n−p)-form, and is defined in a coordinate basis as
?(dxµ1 ∧ · · · ∧dxµp)≡
√
g (n−p)!g
µ1ν1· · ·gµpνp
ν1···νndx
νp+1∧ · · · ∧dxνn,
? ω=
√
g p!(n−p)!g
µ1ν1· · ·gµpνpω
µ1···µpν1···νndx
νp+1 ∧ · · · ∧dxνn,
(52)
whereg ≡detgµν. While the index structure of these relations is clear and natural, the obscure prefactor may be better understood by referring to the equivalent, coordinate-free definition of the Hodge star dual,
ω∧(? σ) =σ∧(? ω) =hω, σivol, (53) for any twop-forms ωandσ. Here the angular brackets indicate a locally defined inner product of the forms, induced by the metric in analogy with the above-defined inner product of vectors,
hω, σi ≡ 1
p!g
µ1ν1· · ·gµpνpωµ
1···µpσν1···νp, (54)
and “vol” stands for the natural volume form on M, associated with the metric,
vol≡√gdx1∧ · · · ∧dxn=?1. (55) The geometric nature of the volume form is made manifest by starting in an orthonormal frame, where one can simply set vol =e∗1∧ · · · ∧e∗n, and then switching to a coordinate basis. Note also that the components of the volume form in a coordinate basis are given by
volµ1...µn =
√
g µ1···µn. (56)
This can be used as a general definition of the Levi-Civita tensor.
The definition (53) now becomes clear. The wedge productω∧(? σ) is a top-dimensional form that is bilinear in ω and σ. The volume form (55) is a natural reference top-dimensional form given the metric, and the bilinearity is ensured by inserting the local inner producthω, σi. The Hodge star operation as defined by Eq. (53) satisfies
? ? ω = (−1)p(n−p)ω, or ?−1 = (−1)p(n−p)? . (57) It is thus self-dual on forms of even degree and on forms of odd degree on odd-dimensional manifolds, and self-dual up to a sign on forms of odd degree on even-dimensional manifolds. The Hodge star dual makes it possible to define a derivative-like operation on differential forms that decreases their degree. This so-called codifferential δ, also sometimes referred to as the
adjoint exterior derivative and denoted as d†, is defined on a p-form ω by
Unlike the exterior derivative d, the codifferential δ is not an antiderivative of the Grassmann algebra. Having now at hand the musical isomorphisms, the Hodge star dual and the d and δ operators, one can define generalizations of other differential operators, familiar from vector calculus. Thus, for instance, one can define thedivergenceof a vector fieldv ∈TMin several different but equivalent ways,
(divv) vol =£vvol, or divv =−δv[ =
1
√
g∂µ(
√
gvµ) =∇µvµ. (59)
The last expression requires an affine connection in addition to the metric, and the Levi-Civita connection is implicitly assumed. The next-to-last expression for divv is a special case of a general formula for the codifferential of ap-form in a coordinate basis,
(δω)µ2···µp =−
1
√
g∂ν1[
√
g(ω])ν1···νp]g
µ2ν2· · ·gµpνp, (ω
])µ1···µp ≡gµ1ν1· · ·gµpνpω
ν1···νp. (60)
Similarly, theLaplace–de Rham operatorgeneralizes the Laplacian to the whole Grassmann algebra of differential forms,
∆≡(d +δ)2 = dδ+δd. (61)
On 0-forms, that is scalar fields (functions), this reduces to ∆f =δdf =−√1
g∂µ(
√
g gµν∂νf) =−gµν∇µ∇νf, (62)
where the last expression again requires the Levi-Civita connection. As a side remark, note that there is another generalization of the Laplacian that can be defined for any tensor field, known as the Laplace–Beltrami operator. The two generalizations of the Laplacian agree with each other on scalar functions, but their action on differential forms differs in a manner specified by the so-called Weitzenb¨ock formula.
Symmetries on manifolds
Intuitively, the concept of a symmetry of a given manifold M seems obvious: this means that the structure of the manifold looks “the same” under some mapping of the manifold to itself. This vague intuition is formalized by the notion of isometry, which is a diffeomorphism φ:M → M that leaves the metric on the manifold invariant,
φ∗g=g. (63)
Given the composition law for pullback, (φ◦χ)∗ =χ∗◦φ∗, all isometries of a given metric form a group. In case that the symmetry is continuous, the one-parametric subgroups corresponding to a set of generators of the symmetry group define a set of flows on the manifold, under which the metric is invariant. The vector fields associated with such flows are calledKilling vectors. A Killing vector ξ can be found by solving any of the following two equivalent conditions,
£ξg= 0, ∇µξν +∇νξµ= 0. (64)
Integration on manifolds
In Euclidean spaces, or on domains therein, integration requires a measure, defined by the line, surface or volume element. Such a measure is not at hand on a general differentiable manifold. To bypass this problem, recall that the (oriented) volume of a parallelepiped in Rn, whose sides are defined by the vectors v1, . . . ,vn, can be calculated as the determinant of the matrix constructed from these vectors, detvµ
a. This determinant is a multilinear map on Rn that has the same symmetries as ann-form. In fact, it can be recovered through the action of a canonical n-form on the vectors, detvµ
a = (dx1 ∧ · · · ∧dxn)(v1, . . . ,vn). This correspondence leads to a natural concept of integration of differential forms on manifolds, defined via the mapping of the forms to Rn through the chosen set of local coordinates.
Consider an n-dimensional manifold M and ann-form defined in terms of local coordinates, ω= 1
n!ωµ1···µndx
µ1 ∧ · · · ∧dxµn =ω
1···ndx1∧ · · · ∧dxn. (65) We would now like to identify dx1∧ · · · ∧dxn with the integration measure dnxand then simply integrate the function ω1···n(x) over the corresponding domain in Rn. This naive definition has
a severe problem though. Namely, the Euclidean volume element dnx is invariant under any permutation of the coordinates, whereas then-form dx1∧ · · · ∧dxn changes sign under any odd permutation. Another manifestation of the same problem is that under a change of coordinates, the function ω1···n(x) is multiplied by the Jacobian of the coordinate transformation, whereas the Euclidean volume element by theabsolute value of the Jacobian. This issue can in principle be bypassed by choosing a fixed order for the coordinates x1, . . . , xn. The problem is that few manifolds possess a globally defined coordinate chart. Even though we would like the value of the integral to be completely independent of the choice of coordinates, we are thus forced to restrict to coordinate redefinitions that do not change the local orientation of the coordinate system, that is have a positive Jacobian. In addition, we have to assume that the manifold is
orientable, that is it possesses an atlas of coordinate charts such that the transition map for any two overlapping charts has a positive Jacobian.
I now define a partition of unity as a set of smooth non-negative functions fi on Mso that
X
i
fi(x) = 1 for all x∈ M. (66)
It is always possible to choose the partition of unity so that only a finite subset of the functions are nonzero at any point x ∈ M, and so that the support of each of fi is covered by a single coordinate chart. This avoids having to deal with infinite sums in the definition of the integral of the n-form ω over M. In fact, it makes it possible to integrate the form fiω using a single chartUi along with its coordinate map ϕi :Ui →Rn. We can then define
Z
M
ω=X i
Z
Ui
fiω, where
Z
Ui fiω≡
Z
ϕi(Ui)
(ϕ−1i )∗ω≡
Z
ϕi(Ui)
ω1···n(x) dnx. (67)
In the last expression, the abstract integral over ann-form onMis pulled back to the domain ϕi(Ui) on Rn. It can be shown that the value of the integral as defined above is independent of the choice of both the atlas of coordinate charts and the partition of unity.
For the latter, an alternative approach to integration of differential forms is more suitable. It is motivated by an intuitive Riemann-like approach to integration in Euclidean spaces, often employed in physics, whereby the integration domain is dissected into small cubes. Since on a general differentiable manifold, one does not have an orthogonal coordinate grid, one instead resorts to a “triangulation” of the manifold, which decomposes the domain of integration into a set of simplexes such that each simplex is covered by a single coordinate chart.
The starting point is the concept of a Euclidean p-simplex,sp ≡(P0, . . . , Pp), defined as the complex envelope of the set of p+ 1 points P0, . . . , Pp ∈Rn, only constrained by the condition that they do not lie in the same (p−1)-dimensional hyperplane. (This is equivalent to requiring that the p vectors, obtained by connecting P0 to Pi, i = 1, . . . , p, are linearly independent.) The next step is to define a vector space, Cp(R), consisting of formal linear combinations
c=X i
cisip, where ci ∈R, (68)
called p-chains. Essentially the same construction can be carried out with the coefficients ci taking values in an arbitrary ring R (the ring of integers Z being a common choice), and the resulting objectCp(R) then has the structure of anR-module. The natural basis ofCp(R) con-sists of allp-simplexes thought of as geometric figures; formally thep-simplex (Pπ(0), . . . , Pπ(p)), whereπ is a permutation of the vertices, is identified with (sgnπ)(P0, . . . , Pp). This gives every simplex a notion of orientation. The oriented boundary of a given simplex can then be identified with the help of theboundary operator. This is a linear map ∂p :Cp →Cp−1, defined by its action on the basis,
∂p(P0, . . . , Pp)≡ p
X
i=0
(−1)i(P0, . . . ,Pˆi, . . . , Pp). (69) It is easy to check that the boundary operator is nilpotent,∂p−1∂p = 0, which is usually phrased colloquially as “a boundary has no boundary.”
Once we have understood the notion of simplexes in Euclidean spaces, we can lift the concept to an arbitrary differentiable manifold M. The standard Euclidean p-simplex ¯sp is defined to have vertices on the positive Cartesian semiaxes, that is, to have Cartesian coordinatesP0µ= 0 for allµandPiµ =δiµfori= 1, . . . , p. For any smooth mapf : ¯sp → M, asingularp-simplex in M, σp, is then defined as the image of ¯sp by the map f, σp ≡ f(¯sp). Singular simplexes in
M are thus in a one-to-one correspondence with smooth maps from the standard Euclidean simplex toM. By means of such smooth maps, all the concepts defined in Euclidean space can be lifted to the manifold M. Thus, a p-chain inM and the boundary operator are defined by
c=X i
ciσip =X i
cifi(¯sp), ∂pσp =f(∂p¯sp), (70)
where the definition of the function f is extended from individual p-simplexes to p-chains by linearity. An integral of ann-formωover ann-chain inMis finally defined by invoking linearity and a pull-back to the standard simplex in Rn,
Z
c ω =ci
Z
σi n
ω, where
Z
σn ω=
Z
¯ sn
f∗ω. (71)
The idea to transfer integration fromMtoRn by using pull-back generalizes to maps between any two manifolds. Indeed, for a smooth map φ :M → N, a differential form ω on N and a chain c inM, one analogously has
Z
φ(c) ω =
Z
c
φ∗ω. (72)
This allows one in particular to perform integration of lower-degree forms over submanifolds such as curved hypersurfaces in Rn, where φ is taken as the corresponding embedding.
With all the definitions at hand, I will now merely state the most important property of integrals of differential forms over manifolds: Stokes’ theorem. Suppose that c is a (p+ 1)-chain and ω a p-form on a manifold M. Then (dropping the subscript on the boundary operator)
Z
c dω =
Z
∂c
ω. (73)
Finally, let me add an important remark that while all the above discussion concerned integra-tion of differential forms, onRiemannian manifolds one can also define integration of functions, which directly generalizes multi-dimensional integration in Euclidean spaces familiar from ele-mentary calculus. This is thanks to the isomorphism of spaces of 0-forms andn-forms, provided by the Hodge star dual,
f 7→? f =f ?1 =fvol, (74) where the volume form vol is given explicitly by Eq. (55). In a plain language more familiar to physicists, an integral of a scalar function on a manifold is geometrically well-defined as long as we use √gdnx as the integration measure in given local coordinates.
Homology and cohomology
The concept of singularp-chains on the manifold defines an algebraic structure onMthat can be used to probe its global topology. One first introduces the chain complex as the sequence of spacesCp(M,R) of singular p-chains along with the mapping between them, defined by the boundary operator,
· · ·−−→∂p+2 Cp+1 ∂p+1
−−→Cp ∂p
−→Cp−1 ∂p−1
−−→ · · · . (75)
Any singular p-chain that has no boundary, that is, is annihilated by the boundary operator ∂p, is called p-cycle. Allp-cycles span a subspace of Cp(M,R), Zp(M,R) = ker∂p. Similarly, any p-chain that is a boundary of a (p+ 1)-chain, that is, can be obtained by acting with the boundary operator ∂p+1 on some (p+ 1)-chain, is called p-boundary. All p-boundaries span a subspace of Cp(M,R),Bp(M,R) = im∂p+1. Due to the nilpotence of the boundary operator, any p-boundary is necessarily a p-cycle, Bp ⊂ Zp. An important result in topology is that while the spaces Zp and Bp are both infinite-dimensional, their quotient is finite-dimensional and characterizes the topology of the manifold. This is the p-th homology group,
Hp(M,R)≡Zp(M,R)/Bp(M,R), 0≤p≤n. (76) Roughly speaking, thep-th homology group consists of all nontrivialp-chains without boundary, defined up to addition of a boundary of a (p+ 1)-chain. Associated with the homology groups are the topological invariants called the Betti numbers,
and theEuler characteristic, which was used by Euler to study convex polyhedra more than two and a half centuries ago,
χ(M)≡
n
X
p=0
(−1)pbp(M). (78)
Homology is relatively easy to visualize but comparatively difficult to compute for concrete manifolds. In order to make it into a practically useful tool, one first defines its dual. This gives rise to the cochain complex,
· · ·←−−dp+1 Cp+1∗ ←−dp Cp∗ ←−−dp−1 Cp−1∗ ←−− · · ·dp−2 , (79) whereCp∗ is the formal dual ofCpas a vector space and dp is the transpose of∂p+1. Likewise, one then defines the space of p-cocyclesasZp(M,
R) = ker dp, and the space of p-coboundaries as Bp(M,R) = im dp−1. The p-th cohomology groupof the manifold M is their quotient,
Hp(M,R)≡Zp(M,R)/Bp(M,R), 0≤p≤n. (80) In more concrete terms, the fact thatCp∗is the dual ofCp means that there is a bilinear mapping (·,·) :Cp×Cp∗ →R that can be used to define the action of cochains on chains,
ω :c7→(c, ω) for any c∈Cp, ω ∈Cp∗. (81) The fact that d is the transpose of ∂ (from now on dropping the subscripts on both operators) is then expressed neatly by the condition
(c,dω) = (∂c, ω). (82)
While the concept of ap-cochain as an element of the dual spaceCp∗ still sounds pretty abstract, the formal analogy of Eq. (82) with the Stokes theorem (73) allows us to construct a concrete realization of the dual structure by identifying p-cochains with p-forms onM and
(c, ω)≡
Z
c
ω. (83)
This defines a homomorphism from the space of p-forms to the space of p-cochains, Cp∗. The coboundary operator d is identified with the exterior derivative. Analogously, the spaceZdRp (M) consists of all closed p-forms ωp such that dωp = 0, whereas the space BdRp (M) consists of all exact p-forms ωp such that there is a (p− 1)-form ωp−1 for which ωp = dωp−1. Their quotient, referred to as the de Rham cohomology, HdRp (M), consists of equivalence classes of closed p-forms defined up to the addition of an exact p-form, and thus classifies topological obstructions to the validity of the Poincar´e lemma on the manifold.
It is not a priori clear how exactly the two different spaces of cochains, cocycles and cobound-aries, introduced above, are related. However, as shown by de Rham, the cohomology groups Hp(M,R) and HdRp (M) are isomorphic provided the manifold M is compact. This is what makes de Rham cohomology so useful, as it allows one to use the full power of exterior calculus to extract topological information about the manifold.
Let us denote as [z]∈Hp the equivalence class of thep-cyclez, and as [ω]∈HdRp the equivalence class of the closedp-formω. Obviously, the integralRzωonly depends on the equivalence classes [z] and [ω], not on the choice of their representative elements, as an immediate consequence of Eq. (82). When restricted to cycles and closed forms, the bilinear mapping defined by Eq. (83) therefore gives rise to a well-defined bilinear form Hp×H
p
Poincar´
e duality
The wedge product of two closed forms is closed and its equivalence class depends only on the equivalence classes of these forms. On closed, compact orientable manifolds, this fact gives rise to a bilinear mapping HdRp ×HdRn−p →R, defined by
[ωp],[ωn−p]7→
Z
M
ωp∧ωn−p for any [ωp]∈HdRp ,[ωn−p]∈H n−p
dR . (84) (Change of the representative elements of the equivalence classes [ωp], [ωn−p] shifts the integrand by an exact form and hence does not modify the value of the integral owing to the fact that the manifold has by assumption no boundary.) This bilinear form is nondegenerate, and hence can be used to define a natural duality between HdRp and HdRn−p, whereby any [ω] ∈ HdRn−p is interpreted as the element of (HdRp )∗ that maps a given [σ]∈HdRp toRMω∧σ.
On the other hand, it follows from our previous discussion that the pairing (83) gives rise to a natural duality between Hp and H
p
dR, which with any p-cycle [z]∈Hp associates an element of (HdRp )∗ that maps a given [σ]∈HdRp toR
zσ. By identifying the two different interpretations of the dual space (HdRp )∗, we arrive at a natural isomorphism betweenHp and H
n−p
dR , which maps a p-cycle [z]∈Hp on the unique equivalence class [ω]∈HdRn−p such that
Z
z σ=
Z
M
ω∧σ for all [σ]∈HdRp . (85)
In loose terms, the (n−p)-formω defines a generalized “Diracδ-function” with support on the cycle z. That this interpretation is sensible is easy to check in special cases, for instance by takingz to be a 0-cycle, that is a point, in which case σis an ordinary function on the manifold and ω a top-dimensional form.
The Poincar´e duality between Hp and H n−p
dR just introduced implies as an immediate corollary the symmetry of the Betti numbers,
bp(M) =bn−p(M). (86) This alone guarantees that the Euler characteristic vanishes on odd-dimensional manifolds. The Poincar´e duality can also be used to study the geometric relationship between cycles of different dimensions on the manifold. Namely, with the mapping [z] 7→ [ωz], defined by Eq. (85), one can introduce yet another bilinear form, Hp×Hn−p →R, by
[zp],[zn−p]7→
Z
M
Hodge theory
By taking advantage of the additional structure provided by Riemannian geometry, one gains additional insight into the importance of cohomology and the link it creates between the topo-logical and the analytical properties of the manifold. We start by introducing an inner product of two p-forms by integrating Eq. (53),
hhω, σii ≡
Z
M
hω, σivol =
Z
M
ω∧(? σ) =
Z
M
σ∧(? ω). (88)
This inner product is symmetric and positive-definite. On a closed, compact orientable man-ifold, it makes the codifferential δ defined by Eq. (58) into the actual adjoint of the exterior derivative operator d. The operator δd now becomes self-adjoint and positive-semidefinite on the Grassmann algebra of M. Its kernel consists of all closed forms. Likewise, the operator dδ becomes self-adjoint and positive-semidefinite and its kernel consists of all coclosed forms (δω = 0). As a consequence, the Laplace–de Rham operator (61) is positive-semidefinite and a given form is harmonic, ∆ω = 0, if and only if it is simultaneously closed and coclosed. By the Hodge decomposition theorem, any p-form can be expressed uniquely as a sum of a closed, coclosed and harmonic component,
ω = dα+δβ+γ, where ∆γ = 0. (89)
The components of the decomposition, dα, δβ and γ, are mutually orthogonal with respect to the inner product (88). This generalizes the Helmholtz decomposition known from elementary vector calculus. The positive-semidefiniteness of the δd and dδ operators implies that if ω is closed then δβ = 0, whereas if ω is coclosed then dα = 0. If ω is neither closed nor coclosed, the individual components can be projected out by acting on ω respectively with dδ and δd,