2.1 Cohomology and geometry of differential forms

Geometric Analysis (Vol II)

ALM 14, pp.129–229 and International Press

Beijing-Boston

Geometric Structures on Riemannian Manifolds

Naichung Conan Leung

Abstract

In this article, we describe various geometries on Riemannian manifolds via a unified approach using normed division algebras and vector cross products. They include K¨ahler geometry, Calabi-Yau geometry, hyperk¨ahler geometry, G2-geometry and geometry of Riemannian symmetric spaces.

Subject Classification: 14D21, 14J32, 32Q25, 53C07, 53C15, 53C25, 53C26, 53C29, 53C38, 53C55, 53D05.

1 Introduction

Many naturally arising Riemannian manifolds admit rich geometric structures.

Projective manifolds in complex algebraic geometry are K¨ahler manifolds. Namely they are Riemannian manifolds with compatible parallel complex structures. K¨ahler manifolds are also the target spaces for two dimensionalσ-models withN = 2 su- persymmetry (SUSY) in physics.

Using the metricg, every orthogonal complex structureJ determines a sym- plectic structureω via

ω(u, v) =g(J u, v) .

Symplectic structures are the natural structures on the phase spaces in classi- cal mechanics. Complex geometry studies objects like complex submanifolds and (stable) holomorphic vector bundles and symplectic geometry studies objects likes (special) Lagrangian submanifolds and unitary flat bundles over them. The mod- ern symplectic geometry incorporates holomorphic curves toquantum correct the classical symplectic geometry.

The mirror symmetry conjecture says roughly that the complex geometry and the quantum symplectic geometry can be transformed to each other between any mirror pair of Calabi-Yau manifolds. Calabi-Yau manifolds are K¨ahler manifolds with C-orientations. Yau proved that every compact K¨ahler manifold with van- ishing first Chern class admits a Calabi-Yau structure. Calabi-Yau manifolds with

∗The Institute of Mathematical Sciences and department of Mathematics, The Chinese Uni- versity of Hong Kong, Shatin, Hong Kong. Email: Leung@math.cuhk.edu.hk

real dimension six arise in string theory as internal spaces whencompactifying ten dimensional spacetimes to our usual spacetimeR^3,1, i.e., writing ten dimensional spacetimes as products of Calabi-Yau manifolds withR^3,1. Similarly, Riemannian manifolds with real dimension 7 (resp. 8) with holonomyG₂(resp. Spin(7)) arise in M-theory (resp. F-theory) as internal spaces.

G₂-manifolds orSpin(7)-manifolds areO-manifolds with or withoutO-orienta- tions. Recall that the octonionOis the largest normed division algebra. These two exceptional geometries can also be interpreted as the geometries of 2-fold or 3-fold vector cross product (VCP). Note that 1-fold VCPs are simply orthogonal complex structures. Namely, K¨ahler manifolds are Riemannian manifolds with parallel 1- fold VCPs. Lagrangian submanifolds and holomorphic curves in K¨ahler manifolds are branes and instantons on VCP-manifolds. Thus we obtain a unified descrip- tion of various geometries of calibrated submanifolds, including (co-)associative submanifolds and Cayley submanifolds.

K¨ahler manifolds withC-VCPs are either Calabi-Yau manifolds or hyperk¨ahler manifolds. ForC-VCPs, instantons depend on a phase angle and there are also two types of branes: D-branes and N-branes, corresponding to Dirichlet and Neumann boundary conditions. Again this gives a unified description of the geometries of special Lagrangian submanifolds, complex hypersurfaces and complex Lagrangian submanifolds.

The space of a certain type of geometric structures modulo symmetries is called amoduli space. For manifolds (resp. bundles), the symmetries are diffeo- morphisms (resp. gauge transformations). A typical example is the moduli space of complex structures.

Some important moduli spacesMhave several complex structures and they are hyperk¨ahler manifolds. Studies of moduli spaces had proven to be a very useful approach to understand the underlying geometries. For instance, moduli spaces play essential roles in Donaldson/Seiberg-Witten theory, Gromov-Witten theory and mirror symmetry.

Note that hyperk¨ahler manifolds areH-orientedH-manifolds. Describing var- ious geometries in terms of normed division algebrasAgives us yet another beau- tiful way to unify many special geometric structures. For instance, when A=C, H and O, (i) A-bundles are holomorphic bundles, B-bundles and Donaldson- ThomasSpin(7)-bundles and (ii) specialA-bundles are Hermitian Yang-Mills bun- dles, hyperholomorphic bundles and Donaldson-Thomas G₂-bundles. Similarly, (i) A/2-Lagrangian submanifolds are Lagrangian submanifolds, hyperlagrangian submanifolds and Cayley submanifolds and (ii) specialA/2-Lagrangian submani- folds are special Lagrangian submanifolds, complex Lagrangian submanifolds and (co-)associative submanifolds. Through these unified descriptions, we have estab- lished new links among different kinds of geometries and uncovered new results from known facts in other geometries.

As a matter of fact, there are not too many other geometric structures on Riemannian manifolds. Accordingly to the Berger’s classification of reduced irre- ducible Riemannian holonomy groups, they must be one of the following: SO(n), U(n),SU(n),Sp(n)Sp(1),Sp(n),Spin(7) orG₂, unlessM is a locally symmet- ric space.

As we will discuss in this survey, these seven possible holonomy groups and their corresponding geometries can be nicely described in terms of normed division algebras (orA-VCPs). Furthermore, using two normed division algebrasAandB, we could also interpret compact Riemannian symmetric spaces as Grassmannians ofcomplex/Lagrangian subspaces in (A⊗B)ⁿ.

In this article, we will first explain the topology and geometry of Riemannian manifolds. Then we explain the complex geometry and the symplectic geometry of K¨ahler manifolds and their mirror symmetry on Calabi-Yau manifolds. After that, we study the geometry of Calabi-Yau threefolds,G₂-manifolds and their dualities.

After we have familiarized ourselves with these geometries, we give two unified approaches to describe them, namely geometry over normed division algebras and VCP-geometry. In the last section, we explain how Riemannian symmetric spaces can also be understood in a similar fashion by using two normed division algebras.

This unification of all geometries of special holonomy will enhance our un- derstanding of each individual geometry and uncover hidden links among them, as well as among different physical theories. There are several excellent sources on the subject of special holonomy, including [13] [23][24][39][59][71][95][114]. Of course, this is a huge subject and the author must apologize that many impor- tant topics are inevitably missed in this article. The choices of topics are due to the personal taste and the limited knowledge of the author. The readers should be warned that the list of references is unfortunately not incomplete as it is not practical to provide explicit references to all the results mentioned in the paper.

2 Topology of manifolds

2.1 Cohomology and geometry of differential forms

Singular and deRham cohomology groups

Inside any given manifold¹ M of dimension n, the most basic cycles are submanifoldsC.In order to allowC to be singular and to have multiplicities, we sometimes use the parametrized version, for example singular cycles.kdimensional singular cycles modulo homologous equivalences form a finitely generated Abelian groupHk(M,Z), thesingular homology.

A dual version of this is thesingular cohomology H^k(M,Z). Modulo torsions, H^k(M,Z) equalsHom(Hk(M,Z),Z). There is a cup product

∪ :H^k(M,Z)⊗H^l(M,Z)→H^k+l(M,Z) ,

giving a ring structure onH^∗(M,Z). SinceM is orientable, we haveHⁿ(M,Z)∼= Zand Poincar´e duality says that∪is a perfect pairing whenk+l=n.

The DeRham theorem says thatH^k(M,R) can be described using differential forms: Let Ω^k(M) be the space of differential forms of degree k, i.e., Ω^k(M) = Γ

M,Λ^kT_M^∗

. We consider the deRham complex

0→Ω⁰(M)→^d Ω¹(M)→ · · ·^d →^d Ωⁿ(M)→0 with d²= 0,

1All manifolds are connected compact oriented smooth manifolds unless specified otherwise.

then

H^k(M,R)∼= Ker(d) Im (d)

Ω^k(M)

.

Moreover the cup product is simply the wedge product of differential forms and the isomorphismHⁿ(M,R)→ Ris given by the integration of the top degree forms.

Volume geometry and symplectic geometry

The existence of a nowhere vanishingn-formvM onM, called thevolume form, is equivalent to the orientability ofM. Note that the existence of a nowhere vanishing 1-form onM is equivalent to the vanishing of the Euler characteristic,χ(M) = 0.

For 2-formsω, the nowhere vanishing should be replaced by the nondegeneracy, i.e., the mapv→ιvω =ω(v,·) gives an isomorphism betweenTM and T_M^∗.This forces M to be of even dimension n = 2m and in this case nondegeneracy is equivalent to the nonvanishing ofω^m/m!, i.e.,ω^m/m! is a volume form onM. A nondegenerate two formω is called a symplectic form on M if it is also a closed form. Symplectic geometry originated from classical mechanics and now it has become an indispensable part of modern mathematics and physics. For higher degree forms, the natural generalizations of nondegeneracy are eitherstable forms, as studied by Hitchin [65] orvector cross product forms (Section 9).

Moser’s lemma says that if two volume forms vM and v_M have the same volume, i.e.,

MvM =

Mv_M, then they are the same up to diffeomorphisms.

Namely, there exists a diffeomorphism f ∈ Dif f⁰(M) satisfying v_M = f^∗vM. Thus the moduli space of volume forms on an oriented manifoldM isR\ {0}. If we require the volume forms to be compatible with a fixed orientation onM, then the moduli space becomesR⁺.

Similarly, if two symplectic forms ωM and ω_M can be connected through symplectic forms representing the same cohomology class inH²(M,R), thenω_M = f^∗ωM for somef ∈Dif f⁰(M). Thus the moduli space of symplectic forms onM islocally given byH²(M,R).

The standard volume form (resp. symplectic form) on Rⁿ is dx¹∧dx²∧

· · · ∧dxⁿ (resp. dx¹∧dx²+dx³∧dx⁴+· · ·+dxⁿ⁻¹∧dxⁿ). In fact every volume (resp. symplectic) form on a manifoldMis locally standard, the so-called Darboux lemma.

We will denote the group of diffeomorphisms ofM preserving a tensort as Dif f(M, t) and its Lie algebra of vector fields asV ect(M, t), i.e.,X ∈V ect(M, t) if and only ifL_Xt = 0. Examples of sucht’s include volume formsv, symplectic formsω and metric tensorsg.

Contracting vector fields with a volume form v_M induces an isomorphism betweenV ect(M) and Ωⁿ⁻¹(M). By the Cartan formula of the Lie derivative on Ω^∗(M),L_X=d◦ι_X+ι_X◦d, we have

LXvM = 0 if and only if d(ιXvM) = 0.

Hence

V ect(M, vM)∼= Ωⁿ⁻¹(M)∩Ker(d) .

Similarly, for a symplectic formωM, we have

V ect(M, ωM)∼= Ω¹(M)∩Ker(d) .

Given any m-dimensional manifold S, the total space of the line bundle Λ^mT_S^∗has a canonical volume formvcan. In terms of any local coordinate system x¹, . . . , x^m, ydx¹∧ · · · ∧dx^m

, we have

vcan=dx¹∧ · · · ∧dx^m∧dy.

Similarly, the total space of the cotangent bundle T_S^∗ has a canonical symplectic formω_can. Actually there is a canonical one formαwithω_can=−dα.Concretely, given any pointp= (x, y)∈T^∗M withy∈T_x^∗M and any tangent vectorvatpwe haveα(v) =y(π_∗v) whereπ:T^∗M →M is the canonical projection (x, y)→x.

In terms of local coordinates

x¹, . . . , x^m, y₁dx¹, . . . , ymdx^m

, we have α=y₁dx¹+· · ·+y_mdx^mand

ωcan=dx¹∧dy₁+· · ·+dx^m∧dym.

The natural class of submanifoldsS in a symplectic manifold (M, ωM) con- sists ofLagrangian submanifolds which are defined as

dimS= 1

2dimM andωM|S= 0.

By an extension of the Darboux lemma, the neighborhood of any Lagrangian sub- manifoldS⊂(M, ωM) is symplectomorphic to a neighborhood ofS⊂(T_S^∗, ωcan).

Furthermore, nearby Lagrangian submanifolds are given by graphs of closed one forms onS.

To rigidify a symplectic manifold (M, ω), we choose a compatible metric g, i.e., if we defineJ :T_M →T_M via

g(J u, v) =ω(u, v),

thenJ is an orthogonal complex structure on each tangent space. J-holomorphic curves Cin M are important tools in modern symplectic geometry. For instance, they play important roles in the Gromov-Witten theory (Section 5.4) and mir- ror symmetry (Section 6.3). Suppose ∂C = φ, then the natural free boundary condition is to require∂C⊂L for a fixed Lagrangian submanifoldL⊂M.

Corresponding notions and results for any volume manifold (M, vM) also hold true and are easier, where Lagrangian submanifolds andJ-holomorphic curves are replaced by hypersurfaces and domains. Indeed they are parts of the theory of the geometry of vector cross products, includingG₂-geometry.

Given any smooth manifold Σ, we consider theevaluation map ev: Σ×M ap(Σ, M)→M

(x, f)→f(x) .

SupposevM is a volume form onM. By integratingev^∗(vM) over Σ, we obtain a differential formω_KΣ(M) onM ap(Σ, M) which descends to the space of knots

K_Σ(M) =M apemb(Σ, M)/Dif f(Σ) .

When Σ has codimension two inM,ω_KΣ(M)is a symplectic form onKΣ(M). We can relate the volume geometry onM with the symplectic geometry onKΣ(M).

Such an interpretation is particularly fruitful when we study the geometry of other vector cross products.

2.2 Hodge theorem

A Riemannian metricgonMis a smooth family of positive definite inner products g_xon tangent spacesT_xM. WhenM is compact and oriented, the Hodge theorem says that we can find a canonical representative for each cohomology class in H^k(M,R), namely the form with the smallest norm. Equivalently,H^k(M,R) can be identified with the orthogonal complement to Im (d) inside Ker(d). That is

H^k(M,R) =Ker(d)∩Ker(d^∗)

=Ker(dd^∗+d^∗d),

whered^∗is the adjoint todand ∆ =dd^∗+d^∗dis a second order elliptic differential operator, called theLaplacian. Hodge theory says that every deRham cohomology class has a unique representativeφsatisfying ∆φ= 0, called aharmonic form. To better understand the Hodge theorem, we need to introduce a few notions.

Riemannian volume forms

SupposeM is oriented, then there is a consistent way to choose the sign of the square root

detgij and define a volume form dvg=

detgijdx¹∧dx²∧ · · · ∧dxⁿ.

We call it theRiemannian volume form of (M, g). Having a volume form allows us to integrate functions onM. In particular vol(M) =

Mdv_g is an important invariant of (M, g). It also allows us to define an inner product on the space of differential forms and other tensors onM as

φ²=

M

|φ(x)|²gdv_g.

and theHodge star operator,∗: Ω^k(M)→Ω^n−k(M) with the defining equation φ∧ ∗η=g(φ, η)dvg.

It satisfies∗²= (−1)^k(n−k).

Letd^∗be the formal adjoint todwith respect to the aboveL²-inner product

·, that is

M

dφ, η_Λk+1T^∗ =

M

φ, d^∗η_ΛkT^∗

for anyφ∈Ω^k(M) andη∈Ω^k+1(M). We have suppressed the volume formdvg

and we will continue to do this. Using· on Ker(d)⊂Ω^k(M), one expects to be able to split the exact sequence

0→Im (d)→Ker(d)→H^k(M)→0 and get

H^k(M)∼=Ker(d)∩(Imd)^⊥

=Ker(d)∩Ker(d^∗)⊂Ω^k(M). By direct computations, we have

d^∗= (−1)^nk+1∗d∗ : Ω^k+1(M)→Ω^k(M) .

We define the second order self-adjoint differential operatorLaplacian as follows,

∆ = (d+d^∗)²=dd^∗+d^∗d: Ω^k(M)→Ω^k(M) .

This operator iselliptic, i.e., its symbol is an invertible algebraic operator. Ellip- ticity is an important notion, for instance its eigenspaces are of finite dimension and the eigenspace decomposition holds on a suititable Hilbert space completion.

For instance, ∆ =−

j

_∂

∂x^j

₂

onRⁿand for functions on a general manifold, we have

∆f = −1

√g

∂

∂xⁱ √

gg^ij ∂f

∂x^j , whereg= detgij.

It is clear that

M

∆φ, φ=

M

|dφ|²+

M

|d^∗φ|².

This implies that ∆ is a self-adjoint (i.e. ∆ = ∆^∗) non-negative operator and Ker(∆) =Ker(d)∩Ker(d^∗) .

The harmonic equation ∆φ= 0 is also the Euler-Lagrange equation for minimizing |φ|²for closed forms representing a fixed cohomology class.

WhenM =S¹ =R/2πZ, we have ∆ =−_dθ^d22. The Fourier series expresses any smooth functionf(θ) =f(θ+ 2π) on S¹as

f(θ) =

ν∈Z

fˆ(ν)e^iνθ.

Note thate^iνθ is an eigenfunction of ∆ with eigenvalueν². The Hodge theorem is a generalization of the Fourier series onS¹ to any compact oriented Riemannian manifold.

TheHodge theorem says that ∆ has an eigenvalue decomposition similar to a symmetric matrix. More precisely (i) each eigenspace

Ω^k_λ(M) :=

φ∈Ω^k(M) : ∆φ=λφ is finite dimensional; (ii) If we denoteSpec(M) =

λ∈R: Ω^k_λ(M)= 0

= λ₀<

λ₁ < · · ·

⊂ R≥0, the spectrum of ∆, then Spec(M) is a unbounded set and without any accumulation point; (iii) Everyφ∈Ω^k(M) can be expressed as

φ= ∞ i=0

φi with φi∈Ω^k_λi uniquely as a convergent power series inL²₁-norm. That is

Ω^k(M)_L2 1 =

∞ i=1

Ω^k_λ

i(M).

The Hodge theorem immediately implies that the dimension of H^k(M) is finite. Since∗: Ω^k(M)→Ω^n−k(M) preserves harmonicity,∗ induces an isomor- phism betweenH^k(M) withH^n−k(M). This gives thePoincar´e duality. However, the wedge product of harmonic forms is usually not harmonic (unless one of them is parallel).

Existence of harmonic representatives for cohomology classes has many im- portant applications. For instance, using the Bochner-Weitzenbock formula, one shows that harmonic one forms on manifolds with positive Ricci curvature must vanish and thereforeH¹(M) = 0 by the Hodge theorem.

The Hodge theorem applies equally well to any elliptic complexes besides the deRham complex. These include the deRham complex coupled with a flat unitary connection; the Dolbeault complex of ¯∂-operator for any complex manifold which can also be coupled with a holomorphic vector bundle and similar elliptic complexes for manifolds with other special geometric structures.

The eigenvalue decomposition of ∆ on Ω^k(M) is compatible with the deRham complex in the sense that d

Ω^k_λ

⊂ Ω^k_λ⁺¹. Hence, for any λ, we have a finite dimensional complex,

0→Ω⁰_λ(M)→^d Ω¹_λ(M)→ · · ·^d →^d Ωⁿ_λ(M)→0,

When λ = 0 this complex is trivial, i.e., d = 0, and otherwise it is an exact complex, i.e.,

(Ker(d)/Im (d))|Ω^∗_λ = 0 forλ= 0.

In particular,

H^k(M)∼= Ker(d) Im (d)

Ω^k(M)

∼= Ω^k₀(M)∼=Ker(∆)|_Ω^k₍M).

The exactness of (Ω^∗_λ, d) implies that

k(−1)^kdim Ω^k_λ= 0 whenλ= 0. Thus χ(M) :=

k

(−1)^kdimH^k(M) =

k

(−1)^k

⎛

⎝

λ≥0

e^−tλdim Ω^k_λ(M)

⎞

⎠

=

λ≥0

e^−tλ

k

(−1)^kdim Ω^k_λ(M)

.

Note that the factors e^−tλ’s are inserted in order to ensure the convergence of the RHS for t > 0 when we interchange the order of summations. By formally lettingtgo to zero, the Euler characteristicχ(M) can be naturally interpreted as

k(−1)^kdim Ω^k(M), the super-dimension of the space of differential forms.

It is interesting to note that for any givenk, dim Ω^k_λ(M) is the multiplicity of the eigenvaluee^−tλ for theheat operator e^−t∆on Ω^k(M). Hence,

χ(M) =

k

(−1)^k

⎛

⎝

λ≥0

e^−tλdim Ω^k_λ(M)

⎞

⎠

=

k

(−1)^kT r_Ωke^−t∆.

The heat operator e^−t∆ has a nice asymptotic expansion as t → 0⁺. Applying some clever tricks, which are motivated from supersymmetry in physics, we can expressχ(M) in terms of the curvature ofM, or characteristic classes. The above arguments are particularly useful in studying the index problem for first order elliptic complexes and give a short proof [5] for the Atiyah-Singer index theorem [6].

2.3 Witten-Morse theory

Suppose f : M → R is a smooth function such that its Hessian ∇²f is non- degenerate at every critical point. Such anf is called aMorse function. Witten [131] studied a twisted version of the deRham complexdt:=e^−tfde^tf usingf for anyt∈R,

0→Ω⁰(M)→^dt Ω¹(M)→ · · ·^dt →^dt Ωⁿ(M)→0 with d²_t = 0.

One can regard this as anon-unitarygauge transformation of the trivial connection onM×Cand hence it defines the same cohomology group, i.e.,

H^k(M)∼= Ker(dt) Im (d_t)

Ω^k(M)

. Notice that the adjoint ofdtis d^∗_t =e^tfd^∗e^−tf and

∆t= (dt+d^∗_t)²

= ∆ +t²|df|²+t∇²f·

Here the action of the Hessian∇²f on differential forms is given by a combination of interior and exterior multiplications

∇²f

ijg^ik _∂

∂xk dx^j∧ .

Astbecomes large, the quadratic term intplays the dominant role. Through semi-classical analysis, one shows that (i) small eigenvalues of ∆tare in one-to-one correspondence with critical points off and (ii) all other eigenvalues go to infinity.

Recall that the cohomology groups can be computed by a truncated finite dimensional complex

Ω^∗_≤C=

λ≤CΩ^∗_λ, dt

for any positive cutoffC for eigen- values. Ast becomes large, one show that

Ω^k_≤C∼=

df(p)=0 index(p)=k

Rp and d_tp=

df(q)=0 index(q)=k+1

n(p, q)q

wheren(p, q) is the (algebraic) number of gradient flow lines off joiningpto q.

Thus H^∗(M) can be computed by counting gradient flow lines between critical points. Therefore Morse theory is just Hodge theory twisted byf!

We remark that the Witten-Morse theory is particularly important in study- ing themiddle dimensionaltopology of various infinite dimensional function spaces.

In these cases, the index of any critical point is infinite but their differences are finite. Examples include the Floer theory for the Chern-Simons functional on the space of connections over three manifolds (Section 2.4); the Floer theory for the symplectic area functional on the loop spaces of symplectic manifolds (Section 5.4) and so on. These structures are parts of topological field theories when we include appropriate boundary theories.

2.4 Vector bundles and gauge theory

A rankrvector bundle overM is a smooth family ofrdimensional vector spaces Exparametrized by x∈M. We write

R^r→E→^π M,

with Ex = π⁻¹(x). The tangent bundle TM is such an example. We can also consider complex vector bundles when each Ex is a complex vector space. By gluing local trivializations of E using a partition of unity, E can always be em- bedded as a subbundle of a trivial bundleM ×C^N. The quotient bundle of the injective homomorphismE →M ×C^N could be regarded as the negative of E.

The construction of integers from natural numbers can be generalized here to give virtual bundle [E₁−E₂] and they generate an Abelian group called thetopological K-theory K(M) ofM.

Given any complex vector bundle E, one can construct cohomology classes which are functorial topological invariants ofE. Examples includes Chern classes c(E) and the Chern charactersch(E). Furthermore,

ch:K(M)→

k=0

H^2k(M,Q)

is a ring homomorphism. Modulo torsions, it is an isomorphism. These charac- teristic classes (modulo torsions) can be described using differential forms via the Chern-Weil theory as described below.

Connections, curvature and Chern classes

First we need to define connections or covariant derivatives, which are first order differential operatorsDA onE-valued differential forms,

DA: Ω⁰(M, E)→Ω¹(M, E) , satisfying

D_A(f s) =df⊗s+f·D_As,

wheref (resp. s) is any function onM (resp. section ofE). We can extendDA

to Ω^k(M, E) by coupling it with d. A sectionsis called parallel, or acovariant constant, if D_As= 0. Locally, E is a trivial bundle andD_A = d+A for some A∈Ω¹(M, End(E)). There are plenty of connections and the difference of any two is an element in Ω¹(M, End(E)). Sinced²= 0,

(DA)²: Ω⁰(M, E)→Ω²(M, E)

can be shown to be a zeroth order operator. Namely there existsF_A∈Ω²(M, End(E)), called thecurvature tensor satisfying

(DA)²s=FAs for any sections. Bianchi identity says that

DAFA= 0∈Ω³(M, End(E)) . This implies that

1 k!T r

i 2πFA

k

∈Ω^2k(M,R)

is a closed differential form for anyk. The cohomology class it represents can be shown to be thek^thChern characterch_k(E)∈H^2k(M,Q). To put them together, we write

ch(E) =r+ n k=1

ch_k(E)

=

exp i

2πF_A

whereris the rank ofE. Similarly the Chern class ofE can be defined as c(E)_R= 1 +c₁(E)_R+· · ·+cr(E)_R

=

det

IE+ i

2πFA ∈

k=0

H^2k(M,R). In particularc₁(E) =ch₁(E) = _i

2πF_A .

Geometrically, a connectionDA identifies any two fibers Ex, Ex along any pathγ(t) joiningxandxinM. This is done by extending any bases₁(x), . . . , sr(x) ofExto the whole path by solving the ODED_γ˙(t)sj(γ(t)) = 0. This identification

is called theparallel transport alongγ. If we choose a polar coordinates around a given pointx₀ ∈ M and use parallel transports along radial paths to identify Exwith Ex₀ for everyxin a small neighborhood U ofx₀, then we obtain a local trivialization

E|U ∼=U×C^r.

With respect to this trivialization, DA coincides with the trivial connection on U×C^rup to first order atx₀, that is

D_A=d+O |x|²

.

Indeed the coefficients of the second order terms are precisely the curvature tensor ofD_A at the pointx₀.

Holonomy and flat connections

The parallel transport along any closed loop based atx₀∈M defines an element inGL(Ex₀), called theholonomy holD_A(γ) ofDAaroundγ. This gives a map on the based loop space ofM,

holD_A : Ωx₀(M)→GL(Ex₀).

The image ofholD_A in GL(r,C) is independent of x₀ ∈M, up to conjugations, called theholonomy group ofDA. Roughly speaking, curvature is the holonomy for infinitesimal loops. In particular, when the curvature is zero,holD_A(γ) equals the identity for any contractible loop. That is holD_A(γ) depends only on the homotopy class ofγ, thus we obtain a homomorphism

holD_A :π₁(M, x₀)→GL(Ex₀) .

Up to conjugacy, this map is independent of the base pointx₀ ∈M. Moreover, if we conjugate DA by an automorphism g ∈ Aut(E) = GE, also called agauge transformation,g·DA=g⁻¹◦DA◦g, we have

F_g·A=g⁻¹F_Ag= 0.

Moreover,DA andg·DA give the same holonomy map up to conjugation. Thus we have a natural identification

{D_A|F_A= 0}/G^hol→Hom(π₁(M), GL(r,C))/Ad(GL(r,C)) . Indeed the universal covering

π₁(M)→M˜ →M

is the universal flat connection is the sense that given any flat connectionDAover M with

ρ:π₁(M)→GL(r,C)

the corresponding holonomy map, then the trivial connectiond on ˜M ×C^r over M˜ descends to the C^r-bundle ˜M ×ρC^r over M and becomes DA, up to gauge symmetries.

The curvature ofDAbeing zero is equivalent to the following sequence 0→Ω⁰(M, E)^D→^AΩ¹(M, E)^D→ · · ·^{A D}→^AΩⁿ(M, E)→0

being a complex, that is (DA)²= 0. Its cohomology group H_D^k_A(M, E) = Ker(DA)

Im (DA)

Ω^k(M,E)

generalizes the deRham cohomology group.

Cohomology groups of low degrees have important geometric significances.

If we denote the symmetry group ofDA as

Aut(E, DA) ={g∈Aut(E) :g·DA=DA},

then its Lie algebra equalsH_D⁰_A(M, End(E))’s. Moreover, the tangent space of a smooth pointD_A in the moduli spaceM^{f lat}(M) of flat connections onE equals H_D¹

A(M, End(E)). This is because ifDA+tB is a flat connection for allB, the flatness (DA+tB)²= 0 is equivalent to

D_AB+t[B, B] = 0.

Modulot, we haveDAB= 0, i.e., [B]∈H_D¹

A(M, End(E)). By similar reasonings, we have the obstructions of smoothness ofM^{f lat}(M) lie insideH_D²

A(M, End(E)).

In order to use the Hodge theorem to represent elements in H^k(M, E) by harmonic forms, we need to choose a metricg onM and also a Hermitian metric honE in such a way thatDAh= 0. As in the manifold case, we have

H_D^k

A(M, E) =Ker(D_AD_A^∗ +D^∗_AD_A)|_Ω^k₍M,E)

whereD_A^∗ is the formal adjoint ofD_A.

Geometrically,D_Ah= 0 means that parallel transports byD_A preserve the inner producth(s₁(x), s₂(x)). Similarly, if we treat our complex vector bundleE as a real vector bundle equipped with a real endomorphismJ :E→E satisfying J²=−IE, thenDAbeing a connection of the complex bundle is equivalent to one on a real bundle which satisfiesDAJ = 0. We could also consider other fiberwise structures on E and connections that preserve them in a similar fashion. Their curvatures will then preserve these structures infinitesimally. For example, for a complex (resp. Hermitian) connection, its curvature lies inside Ω²(M, gl_C(E)) (resp. Ω²(M, u(E))). In general we writeFA∈Ω²(M, ad(E)).

When E is a real vector bundle with a metric g, for instance the tangent bundle of a Riemannian manifold, then the curvature of any orthogonal connection lies inside

F_A∈Ω²(M, ad(E))∼= Ω²

M,Λ²E^∗ becauseso(r)Λ²R^r∗.

Yang-Mills connections

WhenM is an oriented Riemannian manifold, we can measure the length|FA|and define theYang-Mills energy functional,

YM :A(E, h)/GE→R YM(DA) = 1

2

M

|FA|²dvg.

One looks for connections which minimizeYM. The Euler-Lagrange equation for critical points of the Yang-Mills functional is given by the following Yang-Mills equation

D_A^∗F_A= 0∈Ω¹(M, ad(E)) .

This second order system of differential equations in DA is elliptic modulo gauge symmetries. We can usually reduce it to an easier first order equation when M admits special geometric structures. We will see many such examples in this article.

Certainly flat connections are absolute minima for the Yang-Mills functional.

But this is usually an over-determined system of equations unless dim_RM ≤ 3.

When dim_RM = 3, Casson found a way to count the number of flat SU(2)- connections on any homology three spheres and defined theCasson invariant. In this dimension, flat connections are critical points of theChern-Simons functional:

For any connectionDA=d+Aon a topologically trivial bundle, CS(D_A) =

M

T r

A∧dA+2

3A∧A∧A .

This is invariant under gauge transformations in the identity component of GE. ThusCS is only a multi-valued function onA(E)/GE. A more intrinsic way to defineCSis to writeM as the boundaryM =∂Zof a four manifoldZ and extend D_A overZ, then

CS(D_A) =

Z

T r(F_A)².

Floer [42] and others developed the infinite dimensional Witten-Morse theory forCSonA(E)/GE and defined theFloer cohomology HFCS(M) for three man- ifoldsM whose Euler characteristic is the Casson invariant. Roughly speaking, HFCS(M)∼=H^∞/2(A(E)/GE) and it measures the middle dimensional topology ofA(E)/GE [3].

WhenM is the boundary of a four dimensional manifoldZ₁, then theDonald- son polynomial invariants (see Section 4.1) for differentiable structures onclosed four manifolds has a natural generalization forZ₁which takes values inHF_CS(M) and are called the relative Donaldson invariants. Various gluing formulas allow us to compute the Donaldson invariants for a closed four manifoldZ in terms of relative Donaldson invariants for (Zi, M)’s under connected sum decomposition Z=Z₁#MZ₂. This defines a three dimensionaltopological field theory (TFT).

We can further decompose the three manifold M = M₁#_ΣM₂ into a con- nected sum along a Riemann surface Σ of genus g. It is called a handlebody

decomposition if M is a (rational) homology 3-sphere and dimH¹(Mi) = g, i.e.

half ofH¹(Σ) vanish on M₁ and the other half vanish onM₂. The moduli space M^flat_Σ of flatSU(2)-connections on Σ is a symplectic manifold of dimension 6g−6 and eachM^flatM_i is a Lagrangian submanifold inM^flat_Σ . The Lagrangian intersection M^flatM₁ ∩ M^flatM₂ is precisely the discrete set of flat connections on the closed three manifoldM. Thus Atiyah [3] conjectured that HFCS(M) can be identified with another Floer cohomologyHF_Lagr^MflatΣ

M^flatM₁,M^flatM₂

for the Lagrangian intersection theory (Section 5.4).

Witten [133] considered the path integral of Chern-Simons functional over A(E, h)/GE to give a geometric interpretation of the Jones polynomial of knots and also defined new invariants for three manifolds.

A complexified version of this Chern-Simons theory for Calabi-Yau threefolds, and also forG₂-manifolds, plays an important role in mirror symmetry [40][91].

Second fundamental forms

We remark that ifE = E₁⊕E₂ is a direct sum of vector bundles and we write any connectionD onE accordingly

D=

D₁ B₂₁ B₁₂ D₂ ,

then Di is a connection on Ei and we called B₂₁ ∈ Ω¹(M, Hom(E₂, E₁)) and B₁₂∈Ω¹(M, Hom(E₁, E₂)) the second fundamental forms ofD.

IfD is an orthogonal connection with respect to some inner producthonE, then any subbundle E₁ in E determine such a decomposition withE₂ being the orthogonal complement toE₁ andB₂₁=−B₁₂^∗.

3 Riemannian geometry

3.1 Torsion and Levi-Civita connections

SupposeM is a submanifold in R^N, the restriction of the Euclidean metric is a Riemannian metricgonM. Moreover the usual differential is a trivial connection donR^N and it restricts to an orthogonal connection on (M, g).

Given any metricgonM, such an isometric embedding always exists by the celebrated theorem of Nash. There could be many isometric embeddings of a given (M, g) into Euclidean spaces, but the induced orthogonal connections are always the same. It can be characterized intrinsically as the unique connection∇ onTM

satisfying∇g= 0 andT or(∇) = 0, called theLevi-Civita connection of (M, g).

Let us briefly explain the torsion tensorT or(∇) for any connection on the tangent bundle TM. Connections on TM are called affine connections. Each ∇ induces a dual connection onT_M^∗,

∇: Ω⁰(M, T_M^∗)→Ω¹(M, T_M^∗).

Note that (i) Ω⁰(M, T_M^∗)∼= Ω¹(M) and (ii) Ω¹(M, T_M^∗)→^∧ Ω²(M). The difference of ∧ ◦ ∇ : Ω¹(M) → Ω²(M) and the exterior derivatived is given by a tensor,

called the torsion tensor of∇. ExplicitlyT or(∇)∈Ω²(M, TM) is given by T or(∇) (X, Y) =∇XY − ∇YX−[X, Y].

Geometrically, if∇ is any connection onT_M^∗, then using parallel transports along radial paths fromx₀∈M, we obtain a local trivializationU×Rⁿ→ T_U^∗M given by (x;a₁, . . . , a_n)→a₁s¹(x) +· · ·+a_nsⁿ(x) satisfying

∇=d+O |x|²

. Herex=

x¹, ..., xⁿ

are local coordinates onM and

s¹(x), ..., sⁿ(x)

are bases on fibers ofT^∗M. It is natural to require the compatibility that

s¹(x) =dx¹, . . . , sⁿ(x) =dxⁿ. This can be achieved precisely whenT or(∇) = 0.

In such a coordinate system, g =

i,jgij(x)dxⁱ ⊗dx^j has the following property

gij =g ∂

∂xⁱ, ∂

∂x^j =δij+O |x|²

,

for small |x|. Namely gij(0) = δij and dgij(0) = 0. This is called a normal coordinate system and it comes handy when we perform tensor calculus onM.

3.2 Classification of Riemannian holonomy groups

The holonomy group of the Levi-Civita connection of a Riemannian manifold is called the Riemannian holonomy group, or simply the holonomy group and we denote it asHol(M, g). Up to conjugacy, it is a closed subgroup ofO(n). We will only consider the connected component of Hol(M, g). In this article we simply writeHol(M, g) for the identity component, sometimes called the reduced holon- omy group. Every simply connected Riemannian manifold is a Riemannian prod- uct of those with irreducible holonomy groups and they are classified by Berger [11]

as follows: Unless (M, g) is a Riemannian symmetric space, Hol(M, g) must be eitherG_A(n) orH_A(n) for a normed division algebraA,as given in the following table,

A R C H O

G_A(n) O(n) U(n) Sp(n)Sp(1) Spin(7) H_A(n) SO(n) SU(n) Sp(n) G₂ The corresponding geometries are given as follows:

Riemannian K¨ahler Quaternionic-K¨ahlerSpin(7) Volume Calabi-Yau Hyperk¨ahler G₂

The geometry of manifolds with holonomy groupG_A(n) orH_A(n) can be naturally interpreted as the geometry over the normed division algebraAand one with an A-orientation respectively [93] (see section 10).

In section 9, we will also give another unified description of all these geome- tries in terms of real and complex vector cross products. Both approaches are closely related to each other and each has its own advantages.

Riemannian symmetric spaces are classified by Cartan in terms of Lie theory and the list is a bit longer. We will see in section 10 that they can be interpreted as Grassmannians of subspaces in a linear space defined overA⊗Bfor two normed division algebrasAandB.

3.3 Riemannian curvature tensors

The curvature tensorRmof the Levi-Civita connection∇ of (M, g) enjoys many nice properties. The condition∇g= 0 implies thatRm∈Ω²

M,Λ²T_M^∗

and we can view it as a linear operator

Rm: Λ²TM →Λ²TM

called thecurvature operator. This is a symmetric operator because of the torsion freeness of∇. Thus we can writeRm∈Γ

M, Sym²

Λ²T_M^∗ . Explicitly in term of local coordinatexⁱ’s, if we writeg_ij=g _∂

∂x^ι,_∂x^∂_j , then the Levi-Civita connection∇=d+ Γⁱ_jkdx^j⊗dx^k⊗_∂x^∂i is given by the formula

Γ^k_ij =1 2g^kl

∂g_lj

∂xⁱ +∂g_il

∂x^j −∂g_ij

∂x^l .

Here Γ^k_ij’s are called theChristoffel symbols. The Riemannian curvature tensor is given by the formula

−Rmⁱ_qkl=∂Γⁱ_ql

∂x^k −∂Γⁱ_qk

∂x^l + Γⁱ_pkΓ^p_ql−Γⁱ_plΓ^p_qk, whereRm _∂

∂xk,_∂x^∂_{l ∂}

∂x^q =Rmⁱ_qkl∂x^∂_i and we also writeRmpqkl =gpiRmⁱ_qkl. It has the following properties,

0 =Rm_ijkl+Rm_ijlk 0 =Rmijkl−Rmklij

0 =Rmⁱ_jkl+Rmⁱ_klj+Rmⁱ_ljk 0 =Rmijkl,p+Rmijlp,k+Rmijpk,l.

If Rm is identically zero, then the universal cover of M is the Euclidean space Rⁿ with the standard flat metric. More generally, suppose Rm = λI as an operator on Λ²TM, then the universal cover of M must be the sphere or the hyperbolic space according toλbeing a positive or negative constant respectively.