Fundamental Materials of Riemannian Geometry

Transcription

1 Chapter 1 Fundamental aterials of Riemannian Geometry 1.1 Introduction In this chapter, we give fundamental materials in Riemannian geometry. In this book, we assume basic materials on manifolds. We give, for an n-dimensional manifold with Riemannian metric, the several notion of the length of a smooth curve, the distance between two points, Levi-Civita connection, the parallel transport along a curve, geodesics, the curvature tensor fields, integral, the divergence of a smooth vector field, and the Laplace operator, Green s formula, the Laplacian for differential forms, the first and second variational formulas of the length of curves. 1.2 Riemannian anifolds Riemannian metrics Let us recall the definition of an n-dimensional C manifold. A Hausdorff topological space is an n-dimensional C manifold if admits an open covering {U α } α Λ, that is, each U α (α Λ) is an open subset satisfying α Λ U α =, and topological homeomorphisms ϕ α : U α ϕ α (U α ) of open subset U α in onto an open subset ϕ α (U α ) in the n-dimensional Euclidean space R n satisfying that, if U α U β (α, β Λ), ϕ α ϕ β 1 : R n ϕ β (U α U β ) ϕ α (U α U β ) R n is a C diffeomorphism from an open subset ϕ β (U α U β ) in R n onto another open subset ϕ α (U α U β ). A pair (U α, ϕ α ) (α Λ) is called a local chart of. 1

2 2 Spectral Geometry of the Laplacian If (x 1,..., x n ) is the standard coordinate of the n-dimensional Euclidean space R n, for every local chart (U α, ϕ α ), by means of x i α := x i ϕ α (i = 1,..., n), one can define local coordinate (x 1 α,..., x n α) on each open subset U α of. A pair (U α, (x 1 α,..., x n α)) is called local coordinate system. U α U β Figure 1.1 Local charts of. Next, recall the notion of a C Riemannian metric g on an n- dimensional C manifold. Definition 1.1. A C Riemannian metric g on is, by definition, for each point x, g x is a symmetric positive definite bilinear form on the tangent space T x of at x, whose g x is C in x. That is, g x : T x T x R satisfies that: for every u, v, w T x, a, b R, g x (au + bv, w) = a g x (u, w) + b g x (v, w), g x (u, v) = g x (v, u), (1.1) g x (u, u) > 0 (0 u T x ). Then, with respect to local coordinates (U α, (x 1 α,..., x n α)) of, g can be written as g = gij α dx i α dx j α (on U α ). Here, i,j=1 ( gij α = g x i, α x j α ) ( = g x j, α x i α ) = gji. α (1.2) Then, g x is C in x means that, every g α ij is C function on U α.

3 Fundamental aterials of Riemannian Geometry 3 For another local chart (U β, ϕ β ) and local coordinate neighborhood system (U β, (x 1 β,..., xn β )), one can write g = n k,l=1 gβ kl dxk β dxl β, where g β kl = g ( ), x k β x. If Uα U l β, then it holds that, for every β k, l = 1,..., n, g β kl = n g α x i α x j α ij x k i,j=1 β x l β (on U α U β ). (1.3) In fact, since for a C function f : R on, it holds that n k=1 x k β x i α f x k β on U α U β, we have x i α = x k β x i k=1 α x k β f x i α = (i = 1,..., n). (1.4) Substituting this into (1.2), we obtain (1.3). Conversely, if we have (1.3), it holds that dx i x i α α = dx k β (i = 1,..., n), (1.5) x k k=1 β which implies that g = gij α dx i α dx j α = i,j=1 k,l=1 g β kl dxk β dx l β (on U α U β ). (1.6) Therefore, g is determined uniquely independently on a choice of local coordinate neighborhood system (U α, (x 1 α,..., x n α)). Notice that (g α ij ) i,j=1,...,n are C functions on U α whose values are positive definite symmetric matrices of degree n. We denote their determinants by det(g). In the following, we will sometimes denote (U, (x 1,..., x n )) by omitting subscripts α Lengths of curves A continuous curve σ : [a, b] is C 1 curve if, there exists a sufficiently small positive number ɛ > 0 such that, if σ(t), is defined as σ(t) = (σ 1 (t),..., σ n (t)) (t (a ɛ, b + ɛ)) on a local coordinate neighborhood (U, (x 1,..., x n )) around σ(t), each σ i (t) is C 1 function in t on (a ɛ, b+ɛ). Then, one can define the tangent vector σ(t) T σ(t) of a C 1 curve σ(t) by dσ i (t) σ(t) = ( ) x i. (1.7) σ(t)

4 4 Spectral Geometry of the Laplacian Next, if we denote σ(t) 2 := g σ(t) ( σ(t), σ(t)) = i,j=1 dσ i (t) dσ j (t) g ij (σ(t)), (1.8) [a, b] t σ(t) is a continuous function in t, the length L(σ) of a C 1 curve σ : [a, b], can be defined by L(σ) := b a σ(t). (1.9) Now we will discuss the arclength parametrization of a C 1 curve σ : [a, b]. In the following, we always assume that every C 1 curve σ : [a, b] is regular, i.e., σ(t) 0 ( t [a, b]). Then, we can define the length s(t) of the sub-arc σ : [a, t] of a C 1 curve σ by σ(a) s(t) := s(t) t a σ(r) dr. (1.10) σ(t) Figure 1.2 The arclength s(t) of σ. σ(b) Since the differentiation s (t) of s(t) with respect to t is given by s (t) = ds(t) = σ(t) > 0, s(t) is strictly monotone increasing function in t. Thus, one can define its inverse function, by denoting as t = t(s). Therefore, one can define the parametrization in terms of the arclength s of the curve σ by σ(s) := σ(t(s)) (0 s L(σ)). (1.11) If we denote the differentiation of σ with respect to s, by σ (s) and let t (s) := (s) ds, then it holds that σ (s) = dσ (s) (t(s)) ds, and for every s, σ (s) = t (s) dσ (t(s)) = (s) ds(t) = 1. (1.12) ds We usually take the parameter of a C 1 curve σ, a constant multiple of the arclength s, as c s.

5 1.2.3 Distance Fundamental aterials of Riemannian Geometry 5 One can define the distance of a connected C Riemannian manifold (, g) by using the arclength of a C 1 curve: For every two points x, y, let us define d(x, y) := inf{l(σ) σ is a piecewise C 1 curve connecting two points x and y}. (1.13) Here piecewise C 1 curve is a continuous curve connecting a finite number of C 1 curves. Since is arc-wise connected, d(x, y) is finite. Then, d satisfies the three axioms of the distance and (, d) becomes a metric space: (1) d(x, y) = d(y, x) (x, y ), (2) d(x, y) + d(y, z) d(x, z) (x, y, z ), (3) d(x, y) > 0 (x y). d(x, y) = 0 holds if and only if x = y. Furthermore, the topology of a metric space (, d) coincides with the original one which defines a manifold structure of. If the metric space (, d) is complete, i.e., every Cauchy sequence {x k } k=1 of points in, i.e., d(x k, x l ) 0 (k, l ) is convergent. Namely, there exists a point x such that d(x k, x) 0 (k ). We say a Riemannian manifold (, g) is complete if (, d) is so. Every compact Riemannian manifold is complete. We also define the diameter of a compact Riemannian manifold (, g) as 1.3 Connection 0 < diam(, g) := max{d(x, y) x, y } <. (1.14) Levi-Civita connection A vector field X on an n-dimensional C manifold (, g) is X x T x (x ). A C vector field X is, by definition, taking a local coordinate neighborhood system of x, (U α, (x 1 α,..., x n α)) (α Λ), on U α, it can be written as X = n Xi α x, where X i α i C (U α ) (i = 1,..., n, α α Λ). Taking another local coordinate system (U β, (x 1 β,..., xn β )), one write X = n k=1 Xk β on U x k β, it holds that β Xβ k x k β = (on U α U β ; k = 1,..., n), Xα i x i α

6 6 Spectral Geometry of the Laplacian called the changing formula of local coordinates of a vector field X. Now, let us denote by X(), the totality of C vector fields, and by C (), the one of C functions on. For X X() and f C (), Xf C () can be written as (Xf)(x) = n Xi (x) f x (x) (x U) i in terms of local coordinate system (U, (x 1,..., x n )). For every two C vector fields X = n Xi x and Y = n i Y i x X() on, one i can define the third vector field [X, Y ] X() on by [X, Y ] = Then, it holds that { X(Y i ) Y (X i ) } x i = n [X, Y ] f = X(Y f) Y (X f) { n ( X j Y i x j Y j Xi )} x j x i. j=1 (f C ()), where [X, Y ] X() is called the bracket of X and Y. A connection on C (, g) is a C map : X() X() (X, Y ) X Y X() satisfying the following properties: (1) X (Y + Z) = X Y + X Z (2) X+Y Z = X Z + Y Z (3) fx Y = f X Y (4) X (f Y ) = (Xf) Y + f X Y, for f C (), X, Y, Z X(). Then, the following theorem holds. (1.15) Theorem 1.2. Let (, g) be an n-dimensional C Riemannian manifold. One can define a connection, called Levi-Civita connection by the following equation: 2 g( X Y, Z) = X(g(Y, Z)) + Y (g(z, X)) Z(g(X, Y )) + g(z, [X, Y ]) + g(y, [Z, X]) g(x, [Y, Z]), (1.16) for X, Y, Z X(). Then, the Levi-Civita connection satisfies (1) X(g(Y, Z)) = g( X Y, Z) + g(y, X Z), (2) X Y Y X [X, Y ] = 0. Conversely, the only connection satisfying the properties (1) and (2) is the Levi-Civita connection.

7 Fundamental aterials of Riemannian Geometry 7 For every X, Y X(), in the equation (1.16), g(x, Y ) is a C function on defined by g(x, Y )(x) := g x (X x, Y x ) (x ). For the readers, try to prove Theorem 1.2. If we express in terms of local coordinate system (U, (x 1,..., x n )) of, we have x i x j = n Γ k ij k=1 x k as [ x, i x ] = 0, one can obtain j Γ k ij = 1 2 l=1 (where Γ k ij C (U), i, j, k = 1,..., n), g kl ( gjl x i + g il x j g ) ij x l (1.17) for X = x, Y = i x, Z = in (1.16). Here, we denote g j x k ij = g ( ) x, i x, and (g kl ), the inverse matrix of positive definite matrix (g j ij ). is called Christoffel symbol of Levi-Civita connection. Γ k ij Parallel transport For a C 1 curve σ : [a, b] in, X is a C 1 vector field along σ if (1) X(t) T σ(t) ( t [a, b]), and (2) in terms of local coordinates (U, (x 1,..., x n )) at each point σ(t), it holds that ( ) X(t) = ξ i (t) x i T σ(t), where each ξ i (t) is C 1 function in t. Such a vector field X is parallel with respect to connection if σ(t) X = 0. Let σ(t) = (σ 1 (t),..., σ n (t)) be a local expression of a C 1 curve σ. Then, it turns out that the necessary and sufficient condition to hold σ(t) X = 0 is dξ i (t) + j.k=1 Γ i jk(σ(t)) dσj (t) σ(t) ξ k (t) = 0 (i = 1,..., n), (1.18) by means of (1.7) and (1.15). For every C 1 curve σ : [a, b] and an arbitrarily given initial condition of X at x = σ(a), i.e., the coefficients (ξ 1 (a),..., ξ n (a)) of X(a), the parallel vector field X along σ : [a, b] σ(t) X = 0 (a < t < b), ( ) (1.19) X(a) = ξ i (a) x i σ(a)

8 8 Spectral Geometry of the Laplacian X(t) X(b) X(a) σ(a) σ(t) σ(b) Figure 1.3 A vector field along σ. is uniquely determined, because of the existence and uniqueness theorems of the first order ordinary differential system (1.18). In particular, the correspondence P σ : T σ(a) X(a) X(b) T σ(b) is uniquely determined. This correspondence P σ : T σ(a) T σ(b) is a linear isomorphism which satisfies g σ(b) (P σ (u), P σ (v)) = g σ(a) (u, v) (u, v T σ(a) ). (1.20) This is because if we let Y and Z be parallel vector fields along σ with their initial conditions arbitrarily given u, v T σ(a), and X be X(t) = σ(t) (t [a, b]). Then, it holds that d g σ(t)(y (t), Z(t)) = X ( g(y, Z) ) = g( X Y, Z) + g(y, X Z) = 0 since X Y = 0 and X Z = 0. Thus, g σ(t) (Y (t), Z(t)) is constant in t. The correspondence P σ : T σ(a) T σ(b) is called parallel transport along a C 1 curve σ : [a, b] Geodesic A C 1 curve σ : [a, b] in is geodesic if the tangent vector field σ is parallel, i.e., σ(t) σ = 0. In terms of local coordinate system (U, (x 1,..., x n )) of, if we express σ(t) = (σ 1 (t),..., σ n (t)), σ(t) = n dσ i (t) ( x )σ(t) on U, the condition i σ(t) σ = 0 in (1.18) is ξ i (t) = dσi (t) (i = 1,..., n), it holds that d 2 σ i (t) 2 + j,k=1 Γ i jk(σ(t)) dσj (t) dσ k (t) = 0 (i = 1,..., n) (1.21)

9 Fundamental aterials of Riemannian Geometry 9 are the second order ordinary differential system, and for arbitrarily given initial conditions (σ 1 (a),..., σ n (a)) and ( dσ 1 (t) (a),..., dσn (t) (a) ), there exist uniquely solution of (1.21) if t is close enough to a. Namely, for every point p and every vector u T p, there exists a unique geodesic σ(t), passing through p at the initial time and having u as the initial vector at p if t is sufficiently small. Therefore, there exists a unique geodesic satisfying σ(0) = p and σ(0) = u. Let us denote it by σ(t) = Exp p (t u). The exponential map Exp p : T p can be defined locally by T p u σ(1) = Exp p u. It is defined on a neighborhood of 0 in T p. On the problem when the geodesic t Exp p (t u) is extended to < t < for every tangent vector u T p, the following is well known. Theorem 1.3 (Hopf-Rinow). Let (, g) be a connected C Riemannian manifold. Then, the following two conditions are equivalent: (1) (, g) is complete. (2) For every point p, the exponential map Exp p : T p can be defined on the whole space T p. Therefore, in these cases, arbitrarily given two points p and q in can be joined by a geodesic with its length d(p, q). Due to this theorem, for every compact C Riemannian manifold (, g), the exponential map Exp p : T p is defined on the whole space T p. Thus, it is natural to define for every point p, the injectivity radius at p, inj p by inj p := sup{r > 0 Exp p is a diffeomorphism on B r (0 p )}, (1.22) where B r (0 p ) := {u T p g p (u, u) < r 2 } is a ball with radius r, centered at the zero vector 0 p in the tangent space T p at p. Then, we define the injectivity radius of (, g) by inj = inj() := inf{inj p p }. (1.23) For every compact C Riemannian manifold (, g), inj = inj() > 0. Let {v i } n be a basis of T p. Then, the mapping Exp p ( n xi v i ) (x 1,..., x n ) gives a local coordinate system on some neighborhood around p, called normal coordinate system on a neighborhood of p.

10 10 Spectral Geometry of the Laplacian 1.4 Curvature Tensor Fields A tensor field T on of type (r, s) is a C section of the vector bundle r times s times {}}{{}}{ T T T T, namely, if T is expressed in terms of C functions T i1 ir α j 1 j s respect to the local coordinates (U α, (x 1 α,..., x n α)) of, T = T i1 ir αj 1 j s x i1 α x ir α dx j1 α dx js α, on U α, with and it has the same form for other coordinate systems (U β, (x 1 β,..., xn β )), then it holds that, on U α U β ( ), T i1 ir α j 1 j s = T k1 kr β l 1 l s x i1 α x k1 β xir α x kr β x l1 β x j1 α xls β x js α where the right-hand sum is taken over all k 1,..., k r, l 1,..., l s through {1,..., n}. Notice that tensor fields of type (1, 0) are vector fields, and alternating tensor fields of type (0, s) are differential forms of degree s. In terms of Levi-Civita connection of a Riemannian manifold (, g), a tensor field R of type (1, 3) can be defined as follows. For vector fields X, Y, Z X() on, Then, it holds that R(X, Y )Z = X ( Y Z) Y ( X Z) [X,Y ] Z. (1.24) R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0 which is called the first Bianchi identity. Furthermore, it holds that, for α, β, γ C (), R(α X, β Y )(γ Z) = α β γ R(X, Y )Z. (1.25) The tensor field R is called curvature tensor field. Due to (1.25), (R(X, Y )Z) x T x is uniquely determined only on tangent vectors u = X x, v = Y x, w = Z x T x, so that one can write as R(u, v)w = (R(X, Y )Z) x T x. If we write R in terms of local coordinates (U, (x 1,..., x n )) of, as ( R x i, x j ) x k = l=1 R l ijk x l (1 i, j, k n),,

11 Fundamental aterials of Riemannian Geometry 11 it holds that R l ijk = x i Γl kj x j Γl ki + a=1 { } Γ a kj Γ l ai Γ a ki Γ l aj. Taking a linearly independent system {u, v} of the tangent space T x at x of, the quantity K(u, v) := g(r(u, v)v, u) g(u, u) g(v, v) g(u, v) 2 is called the sectional curvature determined by {u, v}. If it holds that K(u, v) > 0 (< 0), for every point x and every linearly independent system {u, v} of T x, then (, g) is called positively curved (negatively curved), respectively. Let {e i } n be an orthonormal basis of (T x, g x ) (x ), one can define a linear map ρ : T x T x by ρ(u) := R(u, e i )e i (u T x ). This linear map is independent of the choice of an orthonormal basis {e i } of T x, and ρ becomes a symmetric tensor field of type (1, 1), called the Ricci transform. The tensor field ρ of type (0, 2) defined by ρ(u, v) = g(ρ(u), v) = g(u, ρ(v)) = g(r(u, e i )e i, v) is called the Ricci tensor. Furthermore, a C function S on defined by S = n ρ(e i, e i ) is called the scalar curvature. These definitions ρ and S are independent of the choice of an orthonormal basis {e i } n. 1.5 Integration Let (, g) be an n-dimensional compact C Riemannian manifold. Let us define the integral f v g of a continuous function f on. For an n-dimensional C Riemannian manifold (, g), let us take a coordinate neighborhood system {(U α, ϕ α ) α Λ} which comes from the manifold structure of. Then, one can give a partition of unity {η α α Λ} subordinate to an open covering {U α } α Λ of. Namely, (i) η α C () (α Λ),

12 12 Spectral Geometry of the Laplacian (ii) 0 η α (x) 1 (x, α Λ), (iii) for each α Λ, the support of η α satisfies supp(η α ) U α, (iv) α Λ η α(x) = 1 (x ). Here, the support of a continuous function f on, supp(f), is by definition the closure of {x f(x) 0}. Now, let us define the integral of a continuous function f whose support is contained in a coordinate neighborhood (U α, (x 1 α,..., x n α)). For such a continuous function f, let us define f v g := (f ϕ 1 α ) det(g) dx 1 α dx n α. (1.26) U α ϕ α(u α) Here, det(g) := det ( g ( )) x, i α x. The differential form j vg of degree n = α dim defined by v g = det(g) dx 1 α dx n α is called the volume form of (, g). Then, for an arbitrary continuous function f on, the integral f v g over is defined by { } f v g = η α f v g = (η α f) v g. (1.27) α Λ α Λ U α Here, the integral U α (η α f) v g over each U α in (1.27) is defined by (1.26) for η α f since supp(η α f) U α. The L 2 inner product (, ) for two continuous functions f and h on, and the L 2 norm of f are defined by (f, h) = f h v g, f = (f, f). The integral of f 1, i.e., Vol(, g) := v g is called the volume of (, g). Since we assume that is compact, it holds that 0 < Vol(, g) <. 1.6 Divergence of Vector Fields and the Laplacian Divergences of vector fields, gradient vector fields and the Laplacian For every C vector field on, X X(), a C function div(x) on, called divergence of a vector field X is defined as follows: Take, first, local coordinates of, (U, (x 1,..., x n )), and orthonormal frame fields {e i } n on U, i.e., T x e i x (x U) satisfies g x (e i x, e j x ) = δ ij. Indeed, {e i } n

13 Fundamental aterials of Riemannian Geometry 13 can be obtained by proceeding the Gram-Schmi orthonormalization to n vector fields { } n x on U which are linearly independent at each point i of U. Then, let X = n Xi x be a local expression of X on U. One i can define div(x) C () by div(x) = g(e i, ei X) = 1 det(g) n ( det(g) X i) x i. (1.28) This definition does not depend on choices of local coordinates (U, (x 1,..., x n )) and local orthonormal frame fields {e i }. For every C function f C () on, one can define the gradient vector field X = grad(f) = f X() is defined by g(y, X) = df(y ) = Y f (Y X()). Taking local coordinates (U, (x 1,..., x n )) on and local orthonormal frame fields {e i } n on U, it holds that grad(f) = f = e i (f) e i = g ij f x j x i, (1.29) i,j=1 where g ij = g ( x i, x j ), and (g kl ) is the inverse matrix of (g ij ). On the cotangent bundle T, a natural metric, denoted by the same symbol g, can be defined in such a way that g(df 1, df 2 ) = g(grad(f 1 ), grad(f 2 )) = g( f 1, f 2 ) (f 1, f 2 C ()). Here, the differential 1-form df Γ(T ) for f C () is defined by df(v) = vf (v T x ), and it holds that df = n f x dx i. i Under these conditions, the Laplacian f C () of every f C () can be defined as follows: f = div(grad f) m 1 = det(g) i,j=1 ( = g ij 2 f n x i x j = i,j=1 ( det(g) g ij f ) x i x j k=1 Γ k ij ) f x k { ei (e i f) ( ei e i )f }. (1.30)

14 14 Spectral Geometry of the Laplacian This linear elliptic partial differential operator : C () f f C () acting on C functions on is called the Laplacian (or the Laplace-Beltrami operator) which depends on a choice of g, so we denote it by g if we want to emphasize it Green s formula We have Proposition 1.4. Let (, g) be an n-dimensional compact Riemannian manifold. For f, f 1, f 2 C () and X X(), (1) f div(x) v g = g(grad(f), X) v g, (2) ( f 1 ) f 2 v g = (3) div(x) v g = 0 g( f 1, f 2 ) v g = (Green s formula). f 1 ( f 2 ) v g, Proof. (1) By ei (f X) = (e i f) X + f ei X, we have div(f X) = g(e i, ei (f X)) = (e i f)g(e i, X) + f g(e i, ei X) = g(grad(f), X) + f div(x). (1.31) Integrate both sides of (1.25) over, by (3), we have 0 = div(f X) v g = g(grad(f), X) v g + f div(x) v g. (2) In (1), let f = f 1, X = grad(f 2 ). Then, we have f 1 ( f 2 ) v g = f 1 div(grad(f 2 )) v g = g(grad(f 1 ), grad(f 2 )) v g, g(grad(f 1 ), grad(f 2 )) = g(grad(f 2 ), grad(f 1 )), g(grad(f 1 ), grad(f 2 )) v g = ( f 1 ) f 2 v g. (3) We use partition of unity 1 = α Λ η α subordinate to an open covering {U α } α Λ of. On each U α, we express X = n Xi α x. Then, i α div(x) v g = div(1 X) v g = div (( ) η α X ) v g = α Λ α Λ div(η α X) v g (finite sum). (1.32)

15 Fundamental aterials of Riemannian Geometry 15 Here, for every α Λ, since supp(η α ) U α, we have div(η α X) v g = div(η α X) v g U α 1 n ( ) det(g) det(g) = ηα U α det(g) x i Xα i dx 1 α dx n α α ( ) det(g) = ηα Xα i dx 1 α dx n α. (1.33) U α x i α Here we get (1.33) = 0. In fact, for each α Λ and i = 1,..., n, the integral ( U α x det(g) i ηα Xα) i dx 1 α α dx n α depends only on the boundary value of det(g) η α Xα i at the boundary of U α, but the boundary value must be 0 since supp(η α ) U α. We completed the proof. 1.7 The Laplacian for Differential Forms In this section, we treat with the Laplacian acting differential forms on an n-dimensional C compact Riemannian manifold (, g). Let us denote by Γ(E), the space of all C sections of a vector bundle E. For every 0 r n, let us define A r () = Γ( r T ) whose elements ω are called r-differential forms on, namely, all ω satisfy ω(x σ(1),..., X σ(r) ) = sgn(σ) ω(x 1,..., X r ) ( σ S r ), and are multilinear maps ω : T T (X }{{} 1,..., X r ) ω(x 1,..., X r ) C (), r times where S r is the permutation group of r letters {1,..., r}, and sgn(σ) is the signature of a permutation σ. Next, we define the exterior differentiation d : A r () A r+1 () by r+1 (dω)(x 1,..., X r+1 ) = ( 1) i+1 X i (ω(x 1,..., X i,, X r+1 )) + i<j( 1) i+j ω([x i, X j ], X 1,..., X i,..., X j,..., X r+1 ) (1.34) for every ω A r () and X 1,..., X r+1 X(). Xi means to delete X i. It is known that d(dω) = 0.

16 16 Spectral Geometry of the Laplacian Thirdly, the Riemannian metric g on induces the natural inner product on the ( n r r) -dimensional linear space Tx (x ), denoted by, x (x ). Then, the L 2 -inner product (, ) on A r () is defined by (ω, η) = ω x, η x x v g (ω, η A r ()). (1.35) {x } Thus, the co-differentiation to the exterior differentiation d : A r () A r+1 (), denoted by δ : A r+1 () A r () is the differential operator which has the property (dω, η) = (ω, δη) (ω A r (), η A r+1 ()). Indeed, the co-differentiation δη A r () (η A r+1 ()) is given by (δη)(x 1,..., X r ) = ( ei η)(e i, X 1,..., X r ), (1.36) for X j X() (j = 1,..., r). Here, {e i } n is a local orthonormal frame field on (, g). The X η (X X()) on the right-hand side (1.36) is the covariant differentiation X ω A r () for differential form ω A r () of degree r which is defined by r ( X ω)(x 1,..., X r ) = X(ω(X 1,..., X r )) ω(x 1,..., X X i,..., X r ). Then, it holds that for ω A r (), r+1 (dω)(x 1,..., X r+1 ) = ( 1) i+1 ( Xi ω)(x 1,..., X i,..., X r+1 ). In particular, if we define for C vector field X X(), a 1-form ω A 1 () by ω(y ) = g(x, Y ) ( Y X()), then we have div(x) = δω. (1.37) Finally, we can define the Laplacian acting on the space A r () of differential forms of degree r by Then, it holds that r := d δ + δ d : A r () A r (). ( r ω, η) = (dω, dη) + (δ ω, δ η) (ω, η A r ()). (1.38) In case of r = 0, A 0 () = C (), for f C (), we have 0 f = δ(df) = ei (df)(e i ) = {e i (e i f) ei e i f} = f. In the following, we always denote = 0.

17 Fundamental aterials of Riemannian Geometry The First and Second Variation Formulas of the Lengths of Curves In this section, we derive the well known first variational formula and the second variational of the length L(c) = b ċ(t) of a a C curve c : [a, b] (, g) in a Riemannian manifold (, g), where ċ(t) = g(ċ(t), ċ(t)) (ċ(t) T c(t) ). Applications of the first and second variational formulas to the eigenvalue problem of the Laplacian will be given in Chap. 3. Given a C curve c : [a, b], its variation is a C mapping α : [a, b] ( ɛ, ɛ) (t, s) α(t, s) for sufficiently small positive number ɛ > 0, which satisfies α(t, 0) = c(t) (t [a, b]). Then, a C family of curves c s : [a, b] is given by c s (t) := α(t, s) (t [a, b]), the mapping α is called deformation (or variation) of c, {c s ɛ < s < ɛ}. Then, for each t [a, b], the tangent vector at s = 0 of a C curve ( ɛ, ɛ) s α(t, s) is given by X(t) := d α(t, s) T c(t) (t [a, b]). s=0 It turns out that X is a C vector field along a curve c. We say X, a variational vector field. Conversely, if a C vector field X (variational vector field) along c is given as X(t) T c(t) (t [a, b]), one can construct a variation α : [a, b] ( ɛ, ɛ) of c by X(t) = d ds α(t, s) T c(t). s=0 For example, we may define α by α(t, s) = Exp c(t) (s X(t)) (s ( ɛ, ɛ), t [a, b]). Now let α : [a, b] ( ɛ, ɛ) be a variation of a C curve c : [a, b], and {c s ɛ < s < ɛ} be a variation of c. Then, let us calculate the first variation of length, d ds L(c s ). s=0

18 18 Spectral Geometry of the Laplacian X(t) X(b) csc(b) X(a) c(a) c(t) Figure 1.4 A variation of a curve c and a variational vector field X. We assume that the parameter t of a curve c = c 0 is a constant multiple of the arclength, and g c(t) (ċ(t), ċ(t)) 1 2 = ċ(t) = l ( t [a, b]). Here, l is a constant independent of t. Then, we have the following theorem. Theorem 1.5 (The first variational formula). d b { d ds L(c s ) = l 1 s=0 a g ( ) ( c(t) X(t), ċ(t) gc(t) X(t), ċ(t) ċ )} } = l {g 1 c(b) (X(b), ċ(b)) g c(a) (X(a), ċ(a)) b l 1 g c(t) (X(t), ċ(t) ċ). (1.39) a Proof. We regard t and s as two vector fields T and V along [a, b] ( ɛ, ɛ), respectively, namely, ( ) ( ) T (t,s) :=, V (t,s) := ((t, s) [a, b] ( ɛ, ɛ)). t (t,s) s (t,s) Then, the tangent vector ċ s of curves c s can be written as ( ) ċ s (t) = c s = α (T ) t in terms of the differentiation α of α. On the other hand, we have X(t) = d ds α(t, s) = α (V ). s=0 s=0

19 Fundamental aterials of Riemannian Geometry 19 Thus, we have d ds L(c s) = = 1 2 b d ds a b a { } g c(t) (ċ s (t), ċ s (t)) 1 2 g c(t) (ċ s (t), ċ s (t)) 1 2 d ds g c(t)(ċ s (t), ċ s (t)). (1.40) Here, notice that d ds g c(t)(ċ s (t), ċ s (t)) = V (t,s) g(α (T ), α (T )) = 2 g( V α (T ), α (T )). (1.41) Here, for X X() and a C map ϕ : N, ϕ (X) means ϕ (X)(x) := ϕ x (X x ) T ϕ(x) N (x ). Then, due to the property (2) of Levi-Civita connection in Theorem 1.2, it holds that V α (T ) T α (V ) α ([V, T ]) = 0, (1.42) and, since [ s, t ] = 0, [V, T ] = 0. Thus, due to (1.42), we have V α (T ) = T α (V ). Substitute this into (1.41), and use the property (1) of Levi-Civita connection in Theorem 1.2, we have (1.41) = 2 g( T α (V ), α (T )) { } = 2 T (g(α (V ), α (T ))) g(α (V ), T α (T )). (1.43) Then, inserting (1.43) into (1.40), (1.40) can be written as follows. (1.40) = b a { g(α (T ), α (T )) 1 2 T (g(α (V ), α (T ))) } g(α (V ), T α (T )). (1.44) Here, putting s = 0, we have α (T ) s=0 = ċ(t), α (V ) s=0 = X(t), T α (T ) s=0 = ċ(t) ċ, so the equation of (1.44) at s = 0 turns out that d b { } d ds L(c s ) = l 1 s=0 a g c(t)(x(t), ċ(t)) g c(t) (X(t), ċ(t) ċ). (1.45) This is the desired equation. We have Theorem 1.5.

20 20 Spectral Geometry of the Laplacian Assume that a C curve c : [a, b] is a geodesic whose parameter t is a constant multiple of the arc length. Then, it is known that the following second variational formula is known (for proof, see [35], Vol. II, p. 81, or [47], p. 124): d 2 b [ ] ds 2 L(c s ) = l 1 g( ċ(t) X, ċ(t) X ) g(r(x, ċ(t))ċ(t), X ) s=0 a b ( ) = l 1 g c(t) ċ(t) ( ċ(t) X ) + R(X, ċ(t))ċ(t), X a + l 1 [g( ċ(t) X, X ) ] t=b, (1.46) t=a where X := X l 1 g(x, ċ(t)) ċ(t), i.e., g(x, ċ(t)) = 0. In particular, if g(x(t), ċ(t)) = 0 (t [a, b]), since X(t) = X (t), we have d 2 b ds 2 L(c s ) = l 1 s=0 a ) g c(t) ( ċ(t) ( ċ(t) X) + R(X, ċ(t))ċ(t), X + l 1 [g( ċ(t) X, X) ] t=b A vector field X(t) along a geodesic c(t) satisfying that is called a Jacobi field.. (1.47) t=a ċ(t) ( ċ(t) X) + R(X, ċ(t))ċ(t) = 0 (1.48)