LECTURE 2:THE RIEMANNIAN METRICAs we have seen last time, in Riemannian geometry there will be lots of sum-mations for quantities with many indices. To simplify notions/computations, fromnowon wewill followtheEinstein Summation Convention: If an expression is a product of several termswith indices, and if an index variable appears twice in this expression, once as anupper index in one term and once as a lower index in another term, then (unlessotherwise stated) the expression is understood to be a summation over all possiblevalues of that index (usually from 1 to the space dimension). For example,ab :=ab,aikbncg :=Zainbrcy.ij,l'Note: an upper index in the denominator will be regarded as a lower index, and vice versaOne should also be aware of how we choose upper and lower indices in this course(trying to meet Einstein summation convention). For example, vector fields arealways indexed by lower indices (like Xi,X2,.-) while the coefficients of vectorfields will be indexed by upper indices (e.g. a'Xi + aX2). Similarly a collectionof 1-forms will be indexed byupper indiceswhile the coefficients of their linearcombinationswill be indexed by lower indices.1.THERIEMANNIANMETRICIDefinition of Riemannian metric.Let M be a smooth manifold of dimension m, in other words, M is a secondcountable Hausdorff topological space such that near every point p E M, there isa neighborhood U of p which is diffeomorphic to a domain in Rm. Moreover, if wedenote by (r,..., &m] the coordinate functions on U, then the tangent space T,Mis spanned by the vectors [Oi, .., Om), and its dual TM (the cotangent space) isspanned by [dr', ... , drm].Definition 1.1. A Riemannian metric g on M is an assignment of an inner product9p(, )= (,-)pon T,M for each p e M, such that the assignment depends smoothly on p.Remarks. (1) As we have seen last time, the Riemannian metric g is motivated bythe first fundamental form of a surface in space. They will be used to measure thelength of curves in M.1
LECTURE 2: THE RIEMANNIAN METRIC As we have seen last time, in Riemannian geometry there will be lots of summations for quantities with many indices. To simplify notions/computations, from now on we will follow the Einstein Summation Convention: If an expression is a product of several terms with indices, and if an index variable appears twice in this expression, once as an upper index in one term and once as a lower index in another terma , then (unless otherwise stated) the expression is understood to be a summation over all possible values of that index (usually from 1 to the space dimension). For example, aib i := X i aib i , aijklb m il cj := X i,j,l a ijklb m il cj . aNote: an upper index in the denominator will be regarded as a lower index, and vice versa. One should also be aware of how we choose upper and lower indices in this course (trying to meet Einstein summation convention). For example, vector fields are always indexed by lower indices (like X1, X2, · · ·) while the coefficients of vector fields will be indexed by upper indices (e.g. a 1X1 + a 2X2). Similarly a collection of 1-forms will be indexed by upper indices while the coefficients of their linear combinations will be indexed by lower indices. 1. The Riemannian metric ¶ Definition of Riemannian metric. Let M be a smooth manifold of dimension m, in other words, M is a second countable Hausdorff topological space such that near every point p ∈ M, there is a neighborhood U of p which is diffeomorphic to a domain in R m. Moreover, if we denote by {x 1 , · · · , xm} the coordinate functions on U, then the tangent space TpM is spanned by the vectors {∂1, · · · , ∂m}, and its dual T ∗ p M (the cotangent space) is spanned by {dx1 , · · · , dxm}. Definition 1.1. A Riemannian metric g on M is an assignment of an inner product gp(·, ·) = ⟨·, ·⟩p on TpM for each p ∈ M, such that the assignment depends smoothly on p. Remarks. (1) As we have seen last time, the Riemannian metric g is motivated by the first fundamental form of a surface in space. They will be used to measure the length of curves in M. 1
2LECTURE2:THERIEMANNIANMETRIC(2)Smooth dependence" if X,Y are two smooth vector fields on an opensubsetU CM, then f(p)=(Xp,Yp)pis a smooth function on U.(3) The Riemannian metric g itself is NOT a metric (aka a distance function) onM. Recall that a distance function on M is a continuous function d : M x M → Rso that for all p,q,r E M,. d(p,q) ≥ 0, and d(p,q) = 0 if and only if p = q;. d(p,q) = d(q,p);. d(p, r) ≤ d(p,q) + d(q,r).However, we will see soon that g induces a natural distance function d on M, andthe topology generated by d on M coincides with its original manifold topology.I Riemannian metric as a tensor field.We may also use the language of tensors.By definitiong: F(TM) ×P(TM) →C(M)defined in the obvious way is Co(M)-bilinear, and thus can be viewed as a (o,2)-tensor on M. The remaining conditions of being an inner product (i.e. symmetricand positive definite)at each point nowbecomes, in the language of tensors, thatthe (O, 2)-tensor g is symmetric and positive definite. So we get another descriptionof a Riemannian metric g:ARiemannian metric g is a smooth symmetric (0,2)-tensor fieldthat is positive definite.We remark that many geometric structures on smooth manifold M are definedasaspecial tensorfield.For example,analmost complexstructureonM isaspecial(1,1)-tensor field, a symplectic structure on M is a special (o,2) tensor field, whilea Poisson structure on M is a special (2, O) tensor field.I Riemannian metric via local coordinates.One can represent the Riemannian metric g using local coordinates as follows.Let {U, r',..., rm be a coordinate patch. We denotegi;(p) = (O, 0;)p.It is easy to see that the functions gi, have the following properties:. For all i, j, gi;(p) is smooth in p.. gij = gii, so the matrix (gi;(p)) is symmetric at any p.The matrix (gi(p)) is also positive definite for any p.Note that although g is intrinsically defined, the functions gij depend on thechoice of coordinate system.If {',...,n] is another coordinated system on U,thenOrk0; =Orrdk
2 LECTURE 2: THE RIEMANNIAN METRIC (2) “Smooth dependence” ⇐⇒ if X, Y are two smooth vector fields on an open subset U ⊂ M, then f(p) = ⟨Xp, Yp⟩p is a smooth function on U. (3) The Riemannian metric g itself is NOT a metric (aka a distance function) on M. Recall that a distance function on M is a continuous function d : M × M → R so that for all p, q, r ∈ M, • d(p, q) ≥ 0, and d(p, q) = 0 if and only if p = q; • d(p, q) = d(q, p); • d(p, r) ≤ d(p, q) + d(q, r). However, we will see soon that g induces a natural distance function d on M, and the topology generated by d on M coincides with its original manifold topology. ¶ Riemannian metric as a tensor field. We may also use the language of tensors. By definition g : Γ∞(TM) × Γ ∞(TM) → C ∞(M) defined in the obvious way is C ∞(M)-bilinear, and thus can be viewed as a (0, 2)- tensor on M. The remaining conditions of being an inner product (i.e. symmetric and positive definite) at each point now becomes, in the language of tensors, that the (0, 2)-tensor g is symmetric and positive definite. So we get another description of a Riemannian metric g: A Riemannian metric g is a smooth symmetric (0, 2)-tensor field that is positive definite. We remark that many geometric structures on smooth manifold M are defined as a special tensor field. For example, an almost complex structure on M is a special (1, 1)-tensor field, a symplectic structure on M is a special (0, 2) tensor field, while a Poisson structure on M is a special (2, 0) tensor field. ¶ Riemannian metric via local coordinates. One can represent the Riemannian metric g using local coordinates as follows. Let {U, x1 , · · · , xm} be a coordinate patch. We denote gij (p) = ⟨∂i , ∂j ⟩p. It is easy to see that the functions gij have the following properties: • For all i, j, gij (p) is smooth in p. • gij = gji, so the matrix (gij (p)) is symmetric at any p. • The matrix (gij (p)) is also positive definite for any p. Note that although g is intrinsically defined, the functions gij depend on the choice of coordinate system. If {x˜ 1 , · · · , x˜ n} is another coordinated system on U, then ˜∂i = ∂xk ∂x˜ i ∂k
3LECTURE2:THERIEMANNIANMETRICItfollowsthatQrkOacl9g := (01,00) =In other words, the matrices (gij) and (gi) are related by the matrix equation(gi) = JT(gij)Jwhere J is the Jacobian matrix whose (i, )-element is ()Since for any smooth vector fields X =Xia, and Y =yia, in U.(Xp,Yp)p=Xi(p)Y(p)(Oi,a,)p = gi(p)X'(p)Y (p)so locallywe can write the 2-tensor g asg = gijdr drj.I The dual Riemannian metric on the cotangent space.Since each matrix (gi;) is positive definite, it is invertible. We will denote by(gi) the inverse matrix of (gij), i.e. they satisfy9igik = k.Then thematrix (gi)is again positive definite, and we can use it to define a dualinner product structure g* on T, M for each p. More explicitly, for any 1-formsw=wdrt and n=nideion U, we defineg*(w,n) = (w, n), := g(p)w;(p)n;(p).We will leave as a simple exercise for the reader to check that this definition isindependent of the choices of coordinates.I The musical isomorphisms.Since g is non-degenerate and bilinear on T,M, it gives us an isomorphismbetween T,M and T,M via!b:T,M →T*M,b(Xp)(Yp) := gp(Xp, Yp).(Pronunciation of b: fat)It is not hard to see that b maps smooth vector fields to smooth 1-forms, and givesrise to a vector bundle isomorphism between TM and T*M.In local coordinates, if we denote X = Xo, and take Y = , for each j, thenb(X)(0) = g(X,0,) = gi,Xi,so we concludeb(x'a.)= gijx'da.1Although dimT,M = dimT,M, without using a Riemannian metric or some other extrastructure, we don't have a natural isomorphism between T,M and T*M
LECTURE 2: THE RIEMANNIAN METRIC 3 It follows that g˜ij := ⟨ ˜∂ i , ˜∂ j ⟩ = ∂xk ∂x˜ i gkl ∂xl ∂x˜ j . In other words, the matrices (˜gij ) and (gij ) are related by the matrix equation (˜gij ) = J T (gij )J where J is the Jacobian matrix whose (i, j)-element is ( ∂xi ∂x˜ j ). Since for any smooth vector fields X = Xi∂i and Y = Y j∂j in U, ⟨Xp, Yp⟩p = X i (p)Y j (p)⟨∂i , ∂j ⟩p = gij (p)X i (p)Y j (p), so locally we can write the 2-tensor g as g = gijdxi ⊗ dxj . ¶ The dual Riemannian metric on the cotangent space. Since each matrix (gij ) is positive definite, it is invertible. We will denote by (g ij ) the inverse matrix of (gij ), i.e. they satisfy gijg jk = δ k i . Then the matrix (g ij ) is again positive definite, and we can use it to define a dual inner product structure g ∗ on T ∗ p M for each p. More explicitly, for any 1-forms ω = ωidxi and η = ηidxi on U, we define g ∗ (ω, η) = ⟨ω, η⟩ ∗ p := g ij (p)ωi(p)ηj (p). We will leave as a simple exercise for the reader to check that this definition is independent of the choices of coordinates. ¶ The musical isomorphisms. Since g is non-degenerate and bilinear on TpM, it gives us an isomorphism between TpM and T ∗ p M via1 ♭ : TpM → T ∗ p M, ♭(Xp)(Yp) := gp(Xp, Yp). (Pronunciation of ♭: flat) It is not hard to see that ♭ maps smooth vector fields to smooth 1-forms, and gives rise to a vector bundle isomorphism between TM and T ∗M. In local coordinates, if we denote X = Xi∂i and take Y = ∂j for each j, then ♭(X)(∂j ) = g(X, ∂j ) = gijX i , so we conclude ♭(X i ∂i) = gijX i dxj . 1Although dim TpM = dim T ∗ p M, without using a Riemannian metric or some other extra structure, we don’t have a natural isomorphism between TpM and T ∗ p M
4LECTURE2:THERIEMANNIANMETRICIn other words, b “lowers the indices" via gij, i.e. changes the coefficients from Xito X, := gijXiWe will denote the inverse map of b by#: T'M →T,M.(Pronunciation of #: sharp)Then in local coordinates,#(w;dr) = gw0,So “raises the indices"via g'i. We will call b and # the musical isomorphisms?Notethatforany1-formwand n,gp(#w, #n) = gijgkiwkgm = jwkng'i = ghwkm = (w,n)p.In other words, the dual inner product g,(w, n) on T, M we mentioned above can bedefined as gp(#w, #n), which is a coordinate-free definition of g*IRiemannian metric for tensors.Given the Riemannian inner product g on T,M and the induced inner productg* on T*M, one may further define a natural inner product T(g), also denoted by gif there is no ambiguity, on the tensor product space (T,M)k(T,M) as follows:Let W=(T,M)× (T*M) (the Cartesian product). Consider themap W × W→R given by((Xi,,Xk,w1,..wt), (Yi,...,Yk,n1,...,n))-g(Xi,Yi)..-g(X,Y)g*(w1,n)...g(wi,n)It is a multi-linear map which is linear in each entry.By universalityof tensor product, it gives rise to a unique bilinear map(T,M)(T,M)1 × (T,M)@(T M)8 -Rwhich can be proven to be an inner product.This inner product can be characterized by the following property: Suppose ei, .:emis an orthonormal basis of (TpM, gp), and el,..:,em its dual basis of (T,M, g,).Then the induced inner product on (TpM)k (T,M)l is defined so that(eir ..@ei Qei @...@ei)form an orthonormal basis.In local coordinates, ifT=Tho.@...@O@dh@@dek and likewisefor a (k,l)-tensor S, thenSai.-ak《T, S)= gibr... gjibigina . igaxTi.+So..As an example, we see that the length square of the metric tensor g itself islgl2=(g,g)=gkggigk=%=m2in music, the symbol b means lower in pitch while the symbol means higher in pitch
4 LECTURE 2: THE RIEMANNIAN METRIC In other words, ♭ “lowers the indices” via gij , i.e. changes the coefficients from Xi to Xi := gijXi . We will denote the inverse map of ♭ by ♯ : T ∗ p M → TpM. (Pronunciation of ♯: sharp) Then in local coordinates, ♯(widxi ) = g ijwi∂j . So ♯ “raises the indices” via g ij . We will call ♭ and ♯ the musical isomorphisms2 . Note that for any 1-form ω and η, gp(♯ω, ♯η) = gijg kiωkg ljηl = δ k j ωkηlg lj = g klωkηl = ⟨ω, η⟩ ∗ p . In other words, the dual inner product g ∗ p (ω, η) on T ∗ p M we mentioned above can be defined as gp(♯ω, ♯η), which is a coordinate-free definition of g ∗ . ¶ Riemannian metric for tensors. Given the Riemannian inner product g on TpM and the induced inner product g ∗ on T ∗M, one may further define a natural inner product T k l (g), also denoted by g if there is no ambiguity, on the tensor product space (TpM) ⊗k ⊗(T ∗ p M) ⊗l as follows: Let W = (TpM) k × (T ∗ p M) l (the Cartesian product). Consider the map W × W → R given by ((X1,· · ·,Xk,ω1,· · ·,ωl),(Y1,· · ·,Yk,η1,· · ·,ηl)) 7→ g(X1,Y1)· · ·g(Xk,Yk)g ∗ (ω1,η1)· · ·g ∗ (ωl ,ηl). It is a multi-linear map which is linear in each entry. By universality of tensor product, it gives rise to a unique bilinear map (TpM) ⊗k ⊗ (T ∗ p M) ⊗l × (TpM) ⊗k ⊗ (T ∗ p M) ⊗l → R which can be proven to be an inner product. This inner product can be characterized by the following property: Suppose e1, · · · , em is an orthonormal basis of (TpM, gp), and e 1 , · · · , em its dual basis of (T ∗ p M, g∗ p ). Then the induced inner product on (TpM) ⊗k ⊗ (T ∗ p M) ⊗l is defined so that {ei1 ⊗ · · · ⊗ eik ⊗ e j1 ⊗ · · · ⊗ e jl} form an orthonormal basis. In local coordinates, if T = T i1···ik j1···jl ∂i1 ⊗ · · · ⊗ ∂ik ⊗ dxj1 ⊗ · · · ⊗ dxjk and likewise for a (k, l)-tensor S, then ⟨T, S⟩ = g j1b1 · · · g jlblgi1a1 · · · gikak T i1···ik j1···jl S a1···ak b1···bl . As an example, we see that the length square of the metric tensor g itself is |g| 2 = ⟨g, g⟩ = g ikg jlgijgkl = δ k j δ j k = m. 2 In music, the symbol ♭ means lower in pitch while the symbol ♯ means higher in pitch
LECTURE2:THERIEMANNIANMETRIC52. RIEMANNIAN MANIFOLDSI Riemannian manifolds: Definition and simplest example.LetMbea smoothmanifold.Definition 2.1. Let g be Riemannian metric on M. Then we call the pair (M,g)a Riemannian manifold. (Sometimes we omit g and say M is a Riemannian manifold.)Erample.The simplest manifold of dimension m is Rm, on which we can endowmany Riemannian metrics:(1) The standard inner product on Rm defines a canonical Riemannian metricgo on Rmviago(X,Y) = xiyiAlternatively, this means the matrix (gi) is the identity matrix:(go)ij= Qij.In the notion of tensors, we can writego= dr @dr +..+darm drm.(2) More generally, for any positive definite mx m matrix A= (aii), the formulagp(Xp, Yp) := X,AYpdefines a Riemannian metric on Rm in which case gt = aij. EquivalentlygA -aidr dr.i.i(3) Since Rm admits a global coordinate system, one may even describe all pos-sible Riemannian metrics on Rm: Endow the space Sym(m) of all m × msymmetric matrices (which is linearly isomorphic to IRm(m+1)/2) the stan-dard smooth structure, then the subset PosSym(m) of all positive definitem × m matrices is open and thus again a smooth manifold. By definition,any smooth mapg : Rm → PosSym(m) C Sym(m)defines a Riemannian metric on Rm,and vice versa.Erample. On the torus Tm = (si)m,one has the following flat Riemannian metricgo=do'dol +...+domdgm.Erample. Consider the upper half plane H? = [(r, y) I y > O). On H? the Riemann-ian metric(dr@dr+dydy)g(r,9)uis known as the Hyperbolic metric, and (H?, g) is known as the hyperbolic plane
LECTURE 2: THE RIEMANNIAN METRIC 5 2. Riemannian manifolds ¶ Riemannian manifolds: Definition and simplest example. Let M be a smooth manifold. Definition 2.1. Let g be Riemannian metric on M. Then we call the pair (M, g) a Riemannian manifold. (Sometimes we omit g and say M is a Riemannian manifold.) Example. The simplest manifold of dimension m is R m, on which we can endow many Riemannian metrics: (1) The standard inner product on R m defines a canonical Riemannian metric g0 on R m via g0(X, Y ) = X i X iY i . Alternatively, this means the matrix (gij ) is the identity matrix: (g0)ij = δij . In the notion of tensors, we can write g0 = dx1 ⊗ dx1 + · · · + dxm ⊗ dxm. (2) More generally, for any positive definite m×m matrix A = (aij ), the formula g A p (Xp, Yp) := X T p AYp defines a Riemannian metric on R m in which case g A ij = aij . Equivalently, g A = X i,j aijdxi ⊗ dxj . (3) Since R m admits a global coordinate system, one may even describe all possible Riemannian metrics on R m: Endow the space Sym(m) of all m × m symmetric matrices (which is linearly isomorphic to R m(m+1)/2 ) the standard smooth structure, then the subset PosSym(m) of all positive definite m × m matrices is open and thus again a smooth manifold. By definition, any smooth map g : R m → PosSym(m) ⊂ Sym(m) defines a Riemannian metric on R m, and vice versa. Example. On the torus T m = (S 1 ) m, one has the following flat Riemannian metric g0 = dθ1 ⊗ dθ1 + · · · + dθm ⊗ dθm. Example. Consider the upper half plane H2 = {(x, y) | y > 0}. On H2 the Riemannian metric g(x,y) = 1 y 2 (dx ⊗ dx + dy ⊗ dy) is known as the Hyperbolic metric, and (H2 , g) is known as the hyperbolic plane
6LECTURE 2:THERIEMANNIAN METRICI Constructing new Riemannian manifolds.There aremany ways to construct new Riemannian manifolds from old ones, forexample,(1) Let (M, gM) and (N, gn) be two Riemannian manifolds, then gM@gn definedby(gM 田 gN)(p.g)((Xp, Yq), (X), Y))) = (gM)p(Xp, X) + (gN)a(Yg, Y))is a Riemannian metric on M x N, whose matrix is simply0((gi)m1xm0(gN)m2xwhere mi, m2 are the dimensions of M and N respectivelyDefinition 2.2. We will call (M × N,gM @ gn) the product Riemannianmanifold of (M,gm)and (N,gn),Forexample,. The Euclidean space (Rm, go) is the Riemannian product of m copies of(R, go).. The torus (Tm, go) is the Riemannian product of m copies of of thestandard circle (s,de@do).The hyperbolic plane (H,g) is NOT a Riemannian products of two 1-dimensional manifolds.(2) Let (N,gn) be a Riemannian manifold, and f : M → N a smooth immersion.ie. df, : T,M - Tr(p)N is injective for all p E M. Then the “pull-backmetric"f*gn on M defined by(f*gN)p(Xp,Yp) = (gN)f(p)(dfp(Xp), dfp(Yp))is a Riemannian metric on MDefinition 2.3.We call gM := f*gn the induced metric or the pulled-backmetric on M with respect to f,and call f :(M,gm)→ (N,gn)an isometricimmersion. If f is an embedding, then f is called an isometric embedding.(3) Let (N, gn) bea Riemannian manifold, and M Nbe an immersed/embeddedsubmanifold.Then the inclusion map t :M →N is an immersion, whichdefines an induced Riemannian metric on M.Definition 2.4. We call (M,t*gn)an immersed/embedded Riemannian sub-manifold of (N, gn). (Usually "Riemannian submanifold" refers to “embedded Rie-mannian submanifold").Note that under theidentification of T,M with dtp(T,M)CT,N, the inducedmetric (t*gn)p, viewed as an inner product or a tensor field, is just therestriction of gn onto the subspace T,M C T,N
6 LECTURE 2: THE RIEMANNIAN METRIC ¶ Constructing new Riemannian manifolds. There are many ways to construct new Riemannian manifolds from old ones, for example, (1) Let (M, gM) and (N, gN ) be two Riemannian manifolds, then gM ⊕gN defined by (gM ⊕ gN )(p,q)((Xp, Yq),(X ′ p , Y ′ q )) = (gM)p(Xp, X′ p ) + (gN )q(Yq, Y ′ q ) is a Riemannian metric on M × N, whose matrix is simply (g1)m1×m1 0 0 (gN )m2×m2 , where m1, m2 are the dimensions of M and N respectively. Definition 2.2. We will call (M × N, gM ⊕ gN ) the product Riemannian manifold of (M, gM) and (N, gN ). For example, • The Euclidean space (R m, g0) is the Riemannian product of m copies of (R, g0). • The torus (T m, g0) is the Riemannian product of m copies of of the standard circle (S 1 , dθ ⊗ dθ). • The hyperbolic plane (H, g) is NOT a Riemannian products of two 1- dimensional manifolds. (2) Let (N, gN ) be a Riemannian manifold, and f : M → N a smooth immersion, i.e. dfp : TpM → Tf(p)N is injective for all p ∈ M. Then the “pull-back metric” f ∗ gN on M defined by (f ∗ gN )p(Xp, Yp) = (gN )f(p)(dfp(Xp), dfp(Yp)) is a Riemannian metric on M. Definition 2.3. We call gM := f ∗ gN the induced metric or the pulled-back metric on M with respect to f, and call f : (M, gM) → (N, gN ) an isometric immersion. If f is an embedding, then f is called an isometric embedding. (3) Let (N, gN ) be a Riemannian manifold, and M ⊂ N be an immersed/embedded submanifold. Then the inclusion map ι : M → N is an immersion, which defines an induced Riemannian metric on M. Definition 2.4. We call (M, ι∗ gN ) an immersed/embedded Riemannian submanifold of (N, gN ). (Usually “Riemannian submanifold” refers to “embedded Riemannian submanifold”). Note that under the identification of TpM with dιp(TpM) ⊂ TpN, the induced metric (ι ∗ gN )p, viewed as an inner product or a tensor field, is just the restriction of gN onto the subspace TpM ⊂ TpN
7LECTURE2:THERIEMANNIANMETRIC(4) Let (M,g) be anyRiemannian manifold, and u:M -→Ran arbitrary smoothfunction on M. Then e2ug defined by(e2"g)p(Xp, Yp) = e2u(P) gp(Xp, Yp)is a Riemannian metric on M.Definition 2.5. We say a Riemannian metric g' on M is conformal to g ifg=e2"gfor some u E C(M).By definition, if two Riemannian metrics g and g are conformal, then whenwe replace g by g', for each p, all vectors in T,M are stretched in length bythe same constant eu(p), while the angle between any pair of vectors in TpMkeeps the same. S? as a Riemannian submanifold of R3Erample. Let M = S? be the unit 2-sphere in R3. The induced Riemannian metricg (from the canonical Riemannian metric go on R3) is known as the round metric.To calculate g locally, we need to choose a coordinate patch.For example, we can use cylindrical coordinates and z to parametrize ?,=V1-2coso, y=V1-z2sino, z=z,with 0<<2π, -1 < z<1. Then-2dr =cosOdz-V1-z2sinadoV1-22andsinodz+V1-22cos Odo.dy=V1-z2Itfollowsgs? = [dada + dydy + dzdzlls?22dzdz+(1-2)dodo+dzdz1-1dzdz+(1-z)dedo.1-~2Alternatively, one may use the colatitude (0,)and the longitude E (0,2)to parametrize s? as=sincosy,y=singsin,z=cosg.A similar computation asabovewill giveusgs = do do + sin? odp @ dp
LECTURE 2: THE RIEMANNIAN METRIC 7 (4) Let (M, g) be any Riemannian manifold, and u : M → R an arbitrary smooth function on M. Then e 2u g defined by (e 2u g)p(Xp, Yp) = e 2u(p) gp(Xp, Yp) is a Riemannian metric on M. Definition 2.5. We say a Riemannian metric g ′ on M is conformal to g if g ′ = e 2u g for some u ∈ C ∞(M). By definition, if two Riemannian metrics g ′ and g are conformal, then when we replace g by g ′ , for each p, all vectors in TpM are stretched in length by the same constant e u(p) , while the angle between any pair of vectors in TpM keeps the same. ¶ S 2 as a Riemannian submanifold of R 3 . Example. Let M = S 2 be the unit 2-sphere in R 3 . The induced Riemannian metric g (from the canonical Riemannian metric g0 on R 3 ) is known as the round metric. To calculate g locally, we need to choose a coordinate patch. For example, we can use cylindrical coordinates θ and z to parametrize S 2 , x = √ 1 − z 2 cos θ, y = √ 1 − z 2 sin θ, z = z, with 0 < θ < 2π, −1 < z < 1. Then dx = −z √ 1 − z 2 cos θdz − √ 1 − z 2 sin θdθ and dy = −z √ 1 − z 2 sin θdz + √ 1 − z 2 cos θdθ. It follows gS2 = [dx ⊗ dx + dy ⊗ dy + dz ⊗ dz]|S2 = z 2 1 − z 2 dz ⊗ dz + (1 − z 2 )dθ ⊗ dθ + dz ⊗ dz = 1 1 − z 2 dz ⊗ dz + (1 − z 2 )dθ ⊗ dθ. Alternatively, one may use the colatitude θ ∈ (0, π) and the longitude φ ∈ (0, 2π) to parametrize S 2 as x = sin θ cos φ, y = sin θ sin φ, z = cos θ. A similar computation as above will give us gS2 = dθ ⊗ dθ + sin2 θdφ ⊗ dφ
8LECTURE 2:THERIEMANNIAN METRICTIsometries and local isometries.Next let's define the notion of"equivalence"in the Riemannian worldDefinition 2.6. Let (M, gM) and (N, gn) be two Riemannian manifolds.(1) If : M -→ N is a local diffeomorphism such that gM = *gn, then we callp a local isometry.(2) If a local isometry : (M, gM) → (N, gn) is invertible, then we call anisometry,in which case we say (M,gm)and (N,gn)are isometric Riemann-ian manifolds.Isometries are crucial in Riemannian geometry since isometric Riemannian man-ifolds will be viewed as the same. Local isometries are also important in studyinglocal invariants like curvatures.Remark. A map : (M,gm) → (N,gn) is a local isometry if and only if for anyp e M, there exists a neighborhood U of p in M so that β : U → p(U) is anisometry.Erample. For any m x m positive definite matrix A, (Rm, g^) is isometric to (Rm, go).[Can you write down the isometry?]Erample. On the set M = R>o × (0, 2), consider the Riemannian metricg = dr @ dr + r2do d.Then themap: M →R2-[(r,0) /≥0),(r,0) -→(r cos9,rsing)(where the latter is endowed with the standard Euclidean metric) is an isometry.[Obviously (M, g) is really the polar coordinate system for R2.]Erample.For the standard metrics onRm and Tm: if weregard Tm =Rm/Zm,thenthe projection π : (Rm,go)→ (Tm, go) is a local isometry but not a global isometry.IThe isometrygroup.Obviously isometries satisfies the following functorality:. If : (M,gm) → (N, gn) is an isometry, the -1 is an isometry.: If : (M, gm) → (N, gn) and : (N, gn) → (P, gp) are two isometries, thenthe composition op: (M,gm)→(P,gp) is again an isometry.In particular, if we letIsom(M, g) = (p : (M,g) → (M,g) / is an isometry)Then Isom(M,g)is agroup.It is a subgroupof thediffeomorphism groupDiff(M) = [ : M -→ M I is a diffeomorphism),Definition 2.7. We call Isom(M, g) the isometry group of (M,g)
8 LECTURE 2: THE RIEMANNIAN METRIC ¶ Isometries and local isometries. Next let’s define the notion of “equivalence” in the Riemannian world. Definition 2.6. Let (M, gM) and (N, gN ) be two Riemannian manifolds. (1) If φ : M → N is a local diffeomorphism such that gM = φ ∗ gN , then we call φ a local isometry. (2) If a local isometry φ : (M, gM) → (N, gN ) is invertible, then we call φ an isometry, in which case we say (M, gM) and (N, gN ) are isometric Riemannian manifolds. Isometries are crucial in Riemannian geometry since isometric Riemannian manifolds will be viewed as the same. Local isometries are also important in studying local invariants like curvatures. Remark. A map φ : (M, gM) → (N, gN ) is a local isometry if and only if for any p ∈ M, there exists a neighborhood U of p in M so that φ : U → φ(U) is an isometry. Example. For any m×m positive definite matrix A, (R m, gA) is isometric to (R m, g0). [Can you write down the isometry?] Example. On the set M = R>0 × (0, 2π), consider the Riemannian metric g = dr ⊗ dr + r 2 dθ ⊗ dθ. Then the map φ : M → R 2 − {(x, 0) | x ≥ 0}, (r, θ) 7→ (r cos θ, r sin θ) (where the latter is endowed with the standard Euclidean metric) is an isometry. [Obviously (M, g) is really the polar coordinate system for R 2 .] Example. For the standard metrics on R m and T m: if we regard T m = R m/Z m, then the projection π : (R m, g0) → (T m, g0) is a local isometry but not a global isometry. ¶ The isometry group. Obviously isometries satisfies the following functorality: • If φ : (M, gM) → (N, gN ) is an isometry, the φ −1 is an isometry. • If φ : (M, gM) → (N, gN ) and ψ : (N, gN ) → (P, gP ) are two isometries, then the composition ψ ◦ φ : (M, gM) → (P, gP ) is again an isometry. In particular, if we let Isom(M, g) = {φ : (M, g) → (M, g) | φ is an isometry}. Then Isom(M, g) is a group. It is a subgroup of the diffeomorphism group Diff(M) = {φ : M → M | φ is a diffeomorphism}. Definition 2.7. We call Isom(M, g) the isometry group of (M, g)
LECTURE 2:THERIEMANNIAN METRIC9For example,. The isometry group of (IRm, go) is the Euclidean group E(m) = O(m) × Rm.. The isometry group of ($?, ground) is the orthogonal group O(3),Remark.A remarkabletheorem proved by Myers and Steenrod in1939 claimsTheorem 2.8 (Myers-Steenrod).Let (M, g) be any Riemannian man-ifold.Then with respect to the compact opentopology,there is asmooth structure on Isom(M,g) so that Isom(M,g) is a Lie group,which is compact if M is compact. Moreover, the obvious action ofIsom(M,g) on M is smooth.On the other hand, as we have learned in the course of smooth manifold, the dif-feomorphism groupDiff(M)can be regarded as an“infinitedimensional Liegroup’whose Lie algebra is the Lie algebra of all smooth vector fields on M (for simplicitywe may assume M is compact). So the isometry group Isom(M,g) of a Riemannianmanifold (M, g), as a Lie group which is finite dimensional and carries a smoothstructure, is much nicer than the diffeomorphism group Diff(M) of the underlyingmanifold M.What is the Lie algebra of Isom(M,g)? They are nothing else butthose vector fields whose flow are isometries, known as Killing vector fields.By the remark above we see that the Riemannian structure is much more rigidthan the smooth structure. As we know, locally manifolds of the same dimensionare always the same. However, this is not the case for Riemannian manifolds: thereare rich local geometry in the Riemannian world.I The existence of Riemannian metric.Thefirstremarkabletheorem in thiscourseisTheorem 2.9. On any smooth manifold M, there erist (many)Riemannian metricson anysmoothmanifold M.We shall give twoproofs of this theorem.The first proof.We first take a locally finite covering of M by coordinate patches[Ua, ra, ... , m]. It is clear that one can choose a Riemannian metric ga on eachUa, e.g. one may takega = dea dra.Let (pa] be a partition of unity subordinate the chosen covering (Ua]. We defineg=paa.Note that this is in fact a finite sum in the neighborhood of each point. It is positivedefinite since for any p e M, there always exist some α such that pa(p)> 0. So it口is a Riemannian metric on M
LECTURE 2: THE RIEMANNIAN METRIC 9 For example, • The isometry group of (R m, g0) is the Euclidean group E(m) = O(m) ⋉ Rm. • The isometry group of (S 2 , ground) is the orthogonal group O(3). Remark. A remarkable theorem proved by Myers and Steenrod in 1939 claims Theorem 2.8 (Myers-Steenrod). Let (M, g) be any Riemannian manifold. Then with respect to the compact open topology, there is a smooth structure on Isom(M, g) so that Isom(M, g) is a Lie group, which is compact if M is compact. Moreover, the obvious action of Isom(M, g) on M is smooth. On the other hand, as we have learned in the course of smooth manifold, the diffeomorphism group Diff(M) can be regarded as an “infinite dimensional Lie group” whose Lie algebra is the Lie algebra of all smooth vector fields on M (for simplicity we may assume M is compact). So the isometry group Isom(M, g) of a Riemannian manifold (M, g), as a Lie group which is finite dimensional and carries a smooth structure, is much nicer than the diffeomorphism group Diff(M) of the underlying manifold M. What is the Lie algebra of Isom(M, g)? They are nothing else but those vector fields whose flow are isometries, known as Killing vector fields. By the remark above we see that the Riemannian structure is much more rigid than the smooth structure. As we know, locally manifolds of the same dimension are always the same. However, this is not the case for Riemannian manifolds: there are rich local geometry in the Riemannian world. ¶ The existence of Riemannian metric. The first remarkable theorem in this course is Theorem 2.9. On any smooth manifold M, there exist (many) Riemannian metrics on any smooth manifold M. We shall give two proofs of this theorem. The first proof. We first take a locally finite covering of M by coordinate patches {Uα, x1 α , · · · , xm α }. It is clear that one can choose a Riemannian metric gα on each Uα, e.g. one may take gα = X i dxi α ⊗ dxi α . Let {ρα} be a partition of unity subordinate the chosen covering {Uα}. We define g = X α ραgα. Note that this is in fact a finite sum in the neighborhood of each point. It is positive definite since for any p ∈ M, there always exist some α such that ρα(p) > 0. So it is a Riemannian metric on M. □
10LECTURE2:THERIEMANNIANMETRICSecond proof. According to the famous Whitney embedding theorem, any smoothmanifold M of dimension m can be embedded into R2m+1 as a smooth submanifold,and thus each Riemannian metric on R2m+1 will induce a Riemannian submanifold口metric on M.Remark. One may ask:How large is the space of all Riemannian metrics on a givensmooth manifold? LetRiem(M)=(g|gisaRiemannianmetricon M)be the set of all Riemannian metrics on M. Motivated by the first proof, it is easyto see that if gi,g2 are two Riemannian metrics on M, so is agi +bg2 for a,b > 0.As a consequence, Riem(M) (as a subset in the infinite dimensional vector space ofall symmetric (0, 2)-tensor fields on M) is a positive convex cone.Of course a natural question is: Given a manifold, can one find a Riemannianmetric that is "best" in some sense? This is one of the main targets in Riemanniangeometry. In this course we shall define various kind of invariants (curvatures) ofRiemannian metrics.and we shall study therelations between these invariants andthe topology of the underlying manifold.Remark. In the second proof we used the Whitney embedding theorem.For Rie-mannian manifolds (M, g), there is a much stronger embedding theorem proved bythe famous Nobel prize (in Economics) winner John Nash in 1956,Theorem 2.10 (Nash embedding theorem).Any m-dimensional Rie-mannian manifold (M, g) can be isometrically embedded into the stan-dard (RN,go) as a Riemannian submanifold, wherem(3m+11)ifMiscompact,Nm(m+1)(3m+11)ifMisnoncompact.For compact manifolds, the dimension N was lowered by Gromov in 1986 tom2+5m+6N2and thenwasfurther lowered byGunther in 1989tom2+5mm2+3m+10N=max22In particular, any 2-dimensional smooth Riemannian manifold can be isometricallyembedded into R10 (instead of R17 by Nash).It is still not known whether thisdimension can be further lowered. 33However, if instead of Comaps (or Cr-maps for r ≥ 3),one only require a "Cl-isometries",ie.Cl-diffeomorphism (β: M -→ N with 'gN = gM, then Nash showed in 1955 that any (Mm,g) canbe Cl-isometrically embedded into R2m+1. Of course "Cl-isometries" are not natural objects inRiemannian geometry,because as wehave seen last time, basicRiemannian geometric quantitieslike curvatures need third order derivatives of the embedding (second order derivatives of theRiemannian metric)
10 LECTURE 2: THE RIEMANNIAN METRIC Second proof. According to the famous Whitney embedding theorem, any smooth manifold M of dimension m can be embedded into R 2m+1 as a smooth submanifold, and thus each Riemannian metric on R 2m+1 will induce a Riemannian submanifold metric on M. □ Remark. One may ask: How large is the space of all Riemannian metrics on a given smooth manifold? Let Riem(M) = {g | g is a Riemannian metric on M} be the set of all Riemannian metrics on M. Motivated by the first proof, it is easy to see that if g1, g2 are two Riemannian metrics on M, so is ag1 + bg2 for a, b > 0. As a consequence, Riem(M) (as a subset in the infinite dimensional vector space of all symmetric (0, 2)-tensor fields on M) is a positive convex cone. Of course a natural question is: Given a manifold, can one find a Riemannian metric that is “best” in some sense? This is one of the main targets in Riemannian geometry. In this course we shall define various kind of invariants (curvatures) of Riemannian metrics, and we shall study the relations between these invariants and the topology of the underlying manifold. Remark. In the second proof we used the Whitney embedding theorem. For Riemannian manifolds (M, g), there is a much stronger embedding theorem proved by the famous Nobel prize (in Economics) winner John Nash in 1956, Theorem 2.10 (Nash embedding theorem). Any m-dimensional Riemannian manifold (M, g) can be isometrically embedded into the standard (R N , g0) as a Riemannian submanifold, where N = m(3m+11) 2 , if M is compact, m(m+1)(3m+11) 2 , if M is noncompact. For compact manifolds, the dimension N was lowered by Gromov in 1986 to N = m2 + 5m + 6 2 , and then was further lowered by G¨unther in 1989 to N = max{ m2 + 5m 2 , m2 + 3m + 10 2 }. In particular, any 2-dimensional smooth Riemannian manifold can be isometrically embedded into R 10 (instead of R 17 by Nash). It is still not known whether this dimension can be further lowered. 3 3However, if instead of C∞ maps (or C r -maps for r ≥ 3), one only require a “C 1 -isometries”, i.e. C 1 -diffeomorphism φ : M → N with φ ∗ gN = gM, then Nash showed in 1955 that any (Mm, g) can be C 1 -isometrically embedded into R 2m+1. Of course “C 1 -isometries” are not natural objects in Riemannian geometry, because as we have seen last time, basic Riemannian geometric quantities like curvatures need third order derivatives of the embedding (second order derivatives of the Riemannian metric)