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