LECTURE1:INTRODUCTION1. RIEMANN'S INAUGURAL LECTUREOn June 10, 1854, B. Riemann gave one of the most famous job talk in the historyof mathematics, with title“On the hypothesis which lie at the foundation ofgeometry".This talk not onlygained a job for him (as a privatdocent at GottingenUniversity), but also offered jobs for many of us including me: two of our courses,Manifolds and Riemannian geometry, born in this probationary inaugural lecture.What Riemann did in this talk was tryingto developahigherdimensionalintrinsic geometry.It is a very broad and abstract generalization of the intrinsicdifferential geometry of surfaces in R3 developed by Gauss'.At the beginning of Riemann's talk was a brief “plan of investigation",in whichhe started with the sentence “geometry presuppose the concept of space". To clearthe confusion over non-Euclidean geometry at that time, he proposed to distinguishmetric properties from the topological properties of the Space.The major part of thetalk was divided into three parts. In part one Riemann introduced the conceptionof manifolds, characterized as locally looks like n-dimensional Euclidean space?.Part two is the major part of the talk, in which Riemann developed the desiredintrinsic geometry, started by introducing a positive definite quadratic form (theRiemannian metric)3 at each point. The crucial question Riemann asked himself inthis part was: when does two Riemannian metrics locally isometric? By a dimensioncounting argument, Riemann argues that there should be a set of n(n- functionswhich will determine the metric completely. They are nothing else but sectionalcurvatures (as a generalization of Gauss curvature for surfaces in R3) associated to'In 1827, Gauss published a famous paper “General investigation of curved surfaces", inwhich he proved his Theorema Egregium ("remarkable theorem" in Latin): the Gauss curvature ofa surface can be determined entirely by measuring distances along paths on the surface (intrinsic)and does not depend on how the surface might be embedded in 3-dimensional space (extrinsic)2Riemann's definition of manifold is a very primitive form.Since most of his audience werenon-mathematicians (faculty of Gottingen University),Riemann tried his best to make his lectureintelligible to general audience. The modern abstract definition of manifolds as"topological spacesthat are Hausdorff, second countable and locallyEuclidean"was introduced by H.Weyl in 1912.3In fact Riemann was also aware of the existence of moregeneral "metrics"that could be usedto measure the length of tangent vectors, including the so-called Finsler metric that was developedbyFinslerin 1918.1
LECTURE 1: INTRODUCTION 1. Riemann’s inaugural lecture On June 10, 1854, B. Riemann gave one of the most famous job talk in the history of mathematics, with title “On the hypothesis which lie at the foundation of geometry”. This talk not only gained a job for him (as a privatdocent at G¨ottingen University), but also offered jobs for many of us including me: two of our courses, Manifolds and Riemannian geometry, born in this probationary inaugural lecture. What Riemann did in this talk was trying to develop a higher dimensional intrinsic geometry. It is a very broad and abstract generalization of the intrinsic differential geometry of surfaces in R 3 developed by Gauss1 . At the beginning of Riemann’s talk was a brief “plan of investigation”, in which he started with the sentence “geometry presuppose the concept of space”. To clear the confusion over non-Euclidean geometry at that time, he proposed to distinguish metric properties from the topological properties of the Space. The major part of the talk was divided into three parts. In part one Riemann introduced the conception of manifolds, characterized as locally looks like n-dimensional Euclidean space2 . Part two is the major part of the talk, in which Riemann developed the desired intrinsic geometry, started by introducing a positive definite quadratic form (the Riemannian metric)3 at each point. The crucial question Riemann asked himself in this part was: when does two Riemannian metrics locally isometric? By a dimension counting argument, Riemann argues that there should be a set of n(n−1) 2 functions which will determine the metric completely. They are nothing else but sectional curvatures (as a generalization of Gauss curvature for surfaces in R 3 ) associated to 1 In 1827, Gauss published a famous paper “General investigation of curved surfaces”, in which he proved his Theorema Egregium (”remarkable theorem” in Latin): the Gauss curvature of a surface can be determined entirely by measuring distances along paths on the surface (intrinsic), and does not depend on how the surface might be embedded in 3-dimensional space (extrinsic). 2Riemann’s definition of manifold is a very primitive form. Since most of his audience were non-mathematicians (faculty of G¨ottingen University), Riemann tried his best to make his lecture intelligible to general audience. The modern abstract definition of manifolds as “topological spaces that are Hausdorff, second countable and locally Euclidean” was introduced by H. Weyl in 1912. 3 In fact Riemann was also aware of the existence of more general “metrics” that could be used to measure the length of tangent vectors, including the so-called Finsler metric that was developed by Finsler in 1918. 1
2LECTURE1:INTRODUCTION2-dimensional vector subspaces of the tangent space! Finally in part three, Riemanndealt withpossible applications, especially to questions in physics.42.RIEMANNIAN GEOMETRYFOR EUCLIDEAN SUBMANIFOLDS:A QUICK SURVEYONUNDERGRADUATEDIFFERENTIALGEOMETRYBefore we introduce the abstract conception of Riemannian metric on a smoothmanifold next time, let's start with some basic geometry that we learned in under-graduate differential geometry course (in a higher dimensional fashion). As one canimagine, differential geometry starts by taking derivative. It turns out that all thoseimportant geometric quantities appears by this way.I Curves in RN.Let : I -→ RN be a smooth curve defined on a finite interval I = [0, T]. Bydefinition the arc length s = s(t) is given bys(t) = //()drSince s is strictly increasing, we may change variable and write as=(s),s E[0, 1],where l is the length ofWe start with the unit tangent vector (s): since ll(s)ll =1, i.e.((s),(s)) = 1,taking derivative one gets("(s),/(s) = 0,i.e. "(s) I(s). In other words, "(s) is a normal vectorBy definition,k(s) := /"(s)IIis called the curvature of at (s), and the vector"(s)n(s) :=/"(s)lis called the principal normal of at (s)Remark. Note that n(s) is again a unit vector. So we may repeat this process.What we will get is the torsion and the binormal. If we continue this process forthe binormal, wewill get Frenet formula.4About 60 years later, Einstein used the theory of pseudo-Riemannian manifolds (a general-ization of Riemannian manifolds) to develop his general theory of relativity. In particular, hisequations for gravitation are constraints on the curvature of spacetime
2 LECTURE 1: INTRODUCTION 2-dimensional vector subspaces of the tangent space! Finally in part three, Riemann dealt with possible applications, especially to questions in physics. 4 2. Riemannian geometry for Euclidean submanifolds: a quick survey on undergraduate differential geometry Before we introduce the abstract conception of Riemannian metric on a smooth manifold next time, let’s start with some basic geometry that we learned in undergraduate differential geometry course (in a higher dimensional fashion). As one can imagine, differential geometry starts by taking derivative. It turns out that all those important geometric quantities appears by this way. ¶ Curves in R N . Let γ : I → R N be a smooth curve defined on a finite interval I = [0, T]. By definition the arc length s = s(t) is given by s(t) = Z t 0 ∥γ ′ (τ )∥dτ. Since s is strictly increasing, we may change variable and write γ as γ = γ(s), s ∈ [0, l], where l is the length of γ. We start with the unit tangent vector γ ′ (s): since ∥γ ′ (s)∥ = 1, i.e. ⟨γ ′ (s), γ′ (s)⟩ = 1, taking derivative one gets ⟨γ ′′(s), γ′ (s)⟩ = 0, i.e. γ ′′(s) ⊥ γ ′ (s). In other words, γ ′′(s) is a normal vector. By definition, κ(s) := ∥γ ′′(s)∥ is called the curvature of γ at γ(s), and the vector n(s) := γ ′′(s) ∥γ ′′(s)∥ is called the principal normal of γ at γ(s). Remark. Note that n(s) is again a unit vector. So we may repeat this process. What we will get is the torsion and the binormal. If we continue this process for the binormal, we will get Frenet formula. 4About 60 years later, Einstein used the theory of pseudo-Riemannian manifolds (a generalization of Riemannian manifolds) to develop his general theory of relativity. In particular, his equations for gravitation are constraints on the curvature of spacetime
3LECTURE 1:INTRODUCTIONI The first fundamental form.Now let M be an n-dimensional manifold embedded into RN.For simplicity wesuppose U C Rn is an open set, and supposeS:UCRn-RNis an injective immersion such that (U)=M (or a portion of M). In what followswe denotea01gik(r)wk5kj,k=1defined on Te(a) M, which is known as the first fundamental form of MI The Second fundamental form.Wemaycontinueto calculatethe second derivativetoget()=(-广(t)+dk dt dt Pi(μ(t),dt2ilj,k=1Whe y(t) - (), Note that th frs tem lie n T(o M. o when pojetingto the normal plane N(t)M = (T(t)M)+, and denotinghik = ProjNe()M(Pjk)
LECTURE 1: INTRODUCTION 3 ¶ The first fundamental form. Now let M be an n-dimensional manifold embedded into R N . For simplicity we suppose U ⊂ R n is an open set, and suppose φ : U ⊂ R n → R N is an injective immersion such that φ(U) = M (or a portion of M). In what follows we denote φj = ∂φ ∂xj , 1 ≤ j ≤ n. Then Tφ(x)M = span(φ1, · · · , φn). Now let µ = (µ 1 , · · · , µn ) : I → U be a curve in U, so that γ = φ ◦ µ : I → M be a curve in R N that sits in M. Then γ ′ (t) = d(φ ◦ µ) dt = Xn j=1 dµj dt φj (µ(t)) The arc length of γ is again given by s(t) = Z t 0 ∥γ ′ (τ )∥dτ. If we denote v j := dµj dt , x = µ(t) and gjk(x) := ⟨φj (x), φk(x)⟩, then we get ds dt 2 = ∥γ ′ (t)∥ 2 = Xn j,k=1 gjk(µ(t))v j v k . After polarizing, we get a quadratic form I(X j v jφj , X k w kφk) := Xn j,k=1 gjk(x)v jw k defined on Tφ(x)M, which is known as the first fundamental form of M. ¶ The Second fundamental form. We may continue to calculate the second derivative to get γ ′′(t) = d 2 (φ ◦ µ) dt2 = Xn j=1 d 2µ j dt2 φj (µ(t)) + Xn j,k=1 dµj dt dµk dt φjk(µ(t)), where φjk(x) = ∂ 2φ ∂xj∂xk (x). Note that the first term lies in Tγ(t)M. So when projecting to the normal plane Nγ(t)M = (Tγ(t)M) ⊥, and denoting hjk = ProjNγ(t)M(φjk)
4LECTURE1:INTRODUCTIONone getshuk=Projn()m("(t).j.k=1For simplicity let's take arc length parametrization, so that w is a unit vector(i.e. Zgk些 = 1), Then we getnE hkuuh = k(s)Projing(o)M(n(s).j,k=1After polarizing,the resulting quadratic formI(Eugi,Ewok) :-Ehkwuk(defined on T,M with value in N,M) is known as the second fundamental form ofM. In the case M is a hypersurface (i.e. n = N-1), by fixing an orientationon M one may identify N(t)M with R, and thus II can be viewed as a real-valuedquadratic form.I The Christoffel symbols.Interesting quantities also appears when we study the tangent component of"(t). Since Pjk(r) - hjk(r) e T(a)M = span(Pi,..., Pn), one may writePik(r)=r'ikpi(r) +hi(r).[=1Paring with thevector Pi,onegets(ik,pi)-Zrijkgu[=1A miracle is that the mysterious coefficients F'jk can be calculated via gik's: FromOugi=《ppi)+《pi,ikonegets(pik, pi)=(Okgij +QigikOigks).So if we denote (gi)=(gi)-1, thenI's=Dg*(gik, p)=g*(0ngg + 0,gk-0.gki),The functions T'ik are known as Christoffel symbols.Note that they are determinedbythefirstfundamental form.In summary,we seethatddp dμkMI"(t)P;(μ(t) mod N()Mdt2dt dtj=1i,k=1
4 LECTURE 1: INTRODUCTION one gets Xn j,k=1 hjkv j v k = ProjNγ(t)M(γ ′′(t)). For simplicity let’s take arc length parametrization, so that Pv jφj is a unit vector (i.e. Pgjk dµj ds dµk ds = 1). Then we get Xn j,k=1 hjkv j v k = κ(s)ProjNγ(s)M(n(s)). After polarizing, the resulting quadratic form II(Xv jφj , Xw kφk) := Xhjkv j v k (defined on TxM with value in NxM) is known as the second fundamental form of M. In the case M is a hypersurface (i.e. n = N − 1), by fixing an orientation on M one may identify Nγ(t)M with R, and thus II can be viewed as a real-valued quadratic form. ¶ The Christoffel symbols. Interesting quantities also appears when we study the tangent component of γ ′′(t). Since φjk(x) − hjk(x) ∈ Tφ(x)M = span(φ1, · · · , φn), one may write φjk(x) = Xn l=1 Γ l jkφl(x) + hjk(x). Paring with the vector φi , one gets ⟨φjk, φi⟩ = Xn l=1 Γ l jkgli. A miracle is that the mysterious coefficients Γl jk can be calculated via gjk’s: From ∂kgij = ⟨φik, φj ⟩ + ⟨φi , φjk⟩ one gets ⟨φjk, φi⟩ = 1 2 (∂kgij + ∂jgik − ∂igkj ). So if we denote (g ij ) = (gij ) −1 , then Γ l jk = X i g il⟨φjk, φi⟩ = 1 2 X i g il(∂kgij + ∂jgik − ∂igkj ). The functions Γl jk are known as Christoffel symbols. Note that they are determined by the first fundamental form. In summary, we see that γ ′′(t) = Xn j=1 d 2µ j dt2 + Xn i,k=1 Γ j ik dµi dt dµk dt ! φj (µ(t)) mod Nγ(t)M
LECTURE 1:INTRODUCTION5I The covariant derivative and geodesics.In particular, f is parametrized by are length s (i.e. gik芸= 1), thenndusdμ'dμkI7Tp;(μ(s) = k(s)Projt.()M(n(s).+ds?dsdsi,k=1j=1The lengthndeuidptidμk>7Kg(s)pi(u(s)ds2dsdsi,k=1L=1is known as the geodesic curvature of . If kg(s) = O, then is called a geodesic.Theyarelocallyshortestpaths(generalizationsof straightlinesinEuclideanspaceand great circles in sphere) in M.More generally, given any vector field X = xi(r)p;(r) along (which, bydefinition,istangent toM everywhere),the samecomputationyieldsdxdpk(ox+rix);(u(t) mod N()M,dtj,k=1which is known as the covariant derivative of the vector field X along I The Riemann curvature.Whataboutvectorfields Nthat arenormal toM?Wemay calculatethetangential component of the derivative of N(r)in a similar way. For thispurposewe write,N(a) =Nk(r)k() mod Ne()M.k=1Start with the equation (N(r), p;(r) = 0. By taking derivative we get(0,N(r), P;(r))+<N,pi)=0,i.e.Nfgkj = -(hig,n).ItfollowsN= -(hij, N)gkjand thusweget,for anynormal vector field Non M,,N(r)=-(hij,N)gipk(r) mod Ne(z)Mj,k
LECTURE 1: INTRODUCTION 5 ¶ The covariant derivative and geodesics. In particular, if γ is parametrized by arc length s (i.e. Pgjk dµj ds dµk ds = 1), then Xn j=1 d 2µ j ds2 + Xn i,k=1 Γ j ik dµi ds dµk ds ! φj (µ(s)) = κ(s)ProjTγ(t)M(n(s)). The length κg(s) := Xn j=1 d 2µ j ds2 + Xn i,k=1 Γ j ik dµi ds dµk ds ! φj (µ(s)) is known as the geodesic curvature of γ. If κg(s) ≡ 0, then γ is called a geodesic. They are locally shortest paths (generalizations of straight lines in Euclidean space and great circles in sphere) in M. More generally, given any vector field X = PXj (x)φj (x) along γ (which, by definition, is tangent to M everywhere), the same computation yields dX dt = Xn j,k=1 ∂kX j + Γj ikX i dµk dt φj (µ(t)) mod Nγ(t)M, which is known as the covariant derivative of the vector field X along γ. ¶ The Riemann curvature. What about vector fields N that are normal to M? We may calculate the tangential component of the derivative of N(x) in a similar way. For this purpose we write ∂iN(x) = Xn k=1 N k i (x)φk(x) mod Nφ(x)M. Start with the equation ⟨N(x), φj (x)⟩ = 0. By taking derivative we get ⟨∂iN(x), φj (x)⟩ + ⟨N, φij ⟩ = 0, i.e. X k N k i gkj = −⟨hij , n⟩. It follows N k i = − X j ⟨hij , N⟩g kj and thus we get, for any normal vector field N on M, ∂iN(x) = − X j,k ⟨hij , N⟩g kjφk(x) mod Nφ(x)M
6LECTURE1:INTRODUCTIONApplying this formula to the normal vector fields hij,we may calculate thetangential component of Pijk=O,o,Okp.SincePij=Er'ijpi+hij,we getPhij=kij=(orF"g+r'ij"k-E(hig,hui)glm)PmmodN(a)Mm=111Since Pkij = Pjik, we get,I"k-"g +(ruI" -I'g,F"u)=(hik, hi)- (hug, hu)glm.[=1[=1We defineRix" :=0,r"i-"+(r'i"-I',")=1and letRijkt :=EglmRijk"m,then we get Ruik = (hik, hjt) - (his, hui). The (0,4)-tensorR(x'ot,Eyii,Ezigj,Zwhkr) :=ZRujkxyiziwkon TM is called the Riemann curuature tensor.It admits many nice symmetryproperties from which one can show that the quantityR(X,Y,X,Y)(X,X)(Y,Y) -(X,Y)2depends only on the two dimensional plane span(X, Y). It is known as the sectionalcurvature of M with respect to the plane. By taking a basis there are n(n-1) suchfunctions, and they are the n(n-) functions first studied by Riemann!
6 LECTURE 1: INTRODUCTION Applying this formula to the normal vector fields hij , we may calculate the tangential component of φijk = ∂i∂j∂kφ. Since φij = PΓ l ijφl + hij , we get φkij = ∂kφij = Xn m=1 ∂kΓ m ij + X l Γ l ijΓ m lk − X l ⟨hij , hkl⟩g lm φm mod Nφ(x)M. Since φkij = φjik, we get ∂jΓ m ik − ∂kΓ m ij + Xn l=1 Γ l ikΓ m lj − Γ l ijΓ m lk = Xn l=1 ⟨hik, hjl⟩ − ⟨hij , hkl⟩ g lm. We define Rijk m := ∂jΓ m ik − ∂kΓ m ij + Xn l=1 Γ l ikΓ m lj − Γ l ijΓ m lk and let Rijkl := X m glmRijk m, then we get Rlijk = ⟨hik, hjl⟩ − ⟨hij , hkl⟩. The (0, 4)-tensor R( XX lφl , XY iφi , XZ jφj , XWkφk) := XRlijkX lY iZ jWk on TxM is called the Riemann curvature tensor. It admits many nice symmetry properties from which one can show that the quantity R(X, Y, X, Y ) ⟨X, X⟩⟨Y, Y ⟩ − ⟨X, Y ⟩ 2 depends only on the two dimensional plane span(X, Y ). It is known as the sectional curvature of M with respect to the plane. By taking a basis there are n(n−1) 2 such functions, and they are the n(n−1) 2 functions first studied by Riemann!