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Laplacians in Geometric Analysis

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LAPLACIANS IN GEOMETRIC ANALYSIS Syafiq Johar syafi[email protected] Contents 1 Trace Laplacian 1 1.1 Connections on Vector Bundles . .1 1.2 Local and Explicit Expressions . .2 1.3 Second Covariant Derivative . .3 1.4 Curvatures on Vector Bundles . .4 1.5 Trace Laplacian . .5 2 Harmonic Functions 6 2.1 Gradient and Divergence Operators . .7 2.2 Laplace-Beltrami Operator . .7 2.3 Harmonic Functions . .8 2.4 Harmonic Maps . .8 3 Hodge Laplacian 9 3.1 Exterior Derivatives . .9 3.2 Hodge Duals . 10 3.3 Hodge Laplacian . 12 4 Hodge Decomposition 13 4.1 De Rham Cohomology . 13 4.2 Hodge Decomposition Theorem . 14 5 Weitzenb¨ock and B¨ochner Formulas 15 5.1 Weitzenb¨ock Formula . 15 5.1.1 0-forms . 15 5.1.2 k-forms . 15 5.2 B¨ochner Formula . 17 1 Trace Laplacian In this section, we are going to present a notion of Laplacian that is regularly used in differential geometry, namely the trace Laplacian (also called the rough Laplacian or connection Laplacian). We recall the definition of connection on vector bundles which allows us to take the directional derivative of vector bundles. 1.1 Connections on Vector Bundles Definition 1.1 (Connection). Let M be a differentiable manifold and E a vector bundle over M. A connection or covariant derivative at a point p 2 M is a map D : Γ(E) ! Γ(T ∗M ⊗ E) 1 with the properties for any V; W 2 TpM; σ; τ 2 Γ(E) and f 2 C (M), we have that DV σ 2 Ep with the following properties: 1. D is tensorial over TpM i.e. DfV +W σ = fDV σ + DW σ: 2. D is R-linear in Γ(E) i.e. DV (σ + τ) = DV σ + DV τ: 3. D satisfies the product rule over Γ(E) i.e. if f 2 C1(M), we have: DV (fσ) = df(V ) · σ + fDV σ: Remark 1.2. Property (1) tells us that (DV σ)p depends on the value of V at p only. Properties (2) and (3) tell us that (DV σ)p depends on the value of σ in a neighbourhood of p. Thus, the connection D gives us a \directional derivative" of the tensor field σ 2 Γ(E) at the point p at the direction V 2 TpM. More technically, the covariant derivative is the infinitesimal difference between the parallel transport of the section σ near p on a curve γ :(−"; ") ! M such that γ(0) = p and γ0(0) = X with the vector σ at p. Due to this, a connection is not necessarily unique on a topological manifold M and it depends on the metric the manifold M is endowed with (since different metrics have different parallel transport associated to it). We can extend this definition to the directional derivative over a section V 2 Γ(TU) on an open set U ⊂ M by doing the derivation pointwise and thus DV σ 2 Γ(EjU ) is a section of E on U. Remark 1.3. Given connections DE and DF on vector bundles E and F respectively, we can define a connection DE⊗F on the tensor bundle of E ⊗ F by: E⊗F E F DX (σ ⊗ µ) = (DX σ) ⊗ µ + σ ⊗ (DX µ): Similarly, a connection DE on vector bundle E induces a connection DE∗ on E∗ by using the product rule and connection on E. Explicitly, if σ 2 E and s 2 E∗, we have the following: E∗ E X(s(σ)) = (DX s)(σ) + s(DX σ): 1 Definition 1.4 (Compatible connection). A connection D on a vector bundle E is compatible with the metric h·; ·i on E if for any X 2 Γ(TM) and σ; µ 2 Γ(E), we have: DX hσ; µi = Xhσ; µi = hDX σ; µi + hσ; DX µi: An obvious example of the connection is the Levi-Civita connection in which we take the vector bundle E to be the tangent bundle TM and the inner product to be the Riemannian metric g. We have the following fundamental theorem of Riemannian geometry: Theorem 1.5 (Fundamental theorem of Riemannian geometry). On each Riemannian manifold (M; g), there exists a unique connection r on TM which is compatible with the metric (i.e. rg ≡ 0) and torsion-free (i.e. rX Y − rY X = [X; Y ] for any vector fields X and Y ). This connection is called the Levi-Civita connection. 1.2 Local and Explicit Expressions Of course, the above notations are nice but computation wise, it can be quite limited. Therefore, we can introduce a moving frame to make things more explicit. Definition 1.6. A moving frame for a vector bundle is a collection of n vector fields fv1(x); v2(x); : : : ; vn(x)g ⊂ Γ(TM) such that at each x 2 M, this set forms a basis for TxM. There are many useful frames that can be used in computation. For example, if we have a local coordinate patch U ⊂ M given by (x1; x2; : : : ; xn), we can define a coordinate frame @ @ @ @x1 ; @x2 ;:::; @xn which forms a basis for TxM for all x 2 U. For example of a local compu- @ @ tation, we have, g @xi ; @xj = gij. Another useful choice of frames is the orthonormal frame which is defined independently of coordinates. The orthonormal frame is the collection fe1(x); e2(x); : : : ; en(x)g such that g(ei; ej) = δij. Note here that the orthonormal frame does not define a coordinate system on M locally (since, in general, [ei; ej] 6= 0 but the Lie bracket of coordinate vectors vanish i.e @ @ @xi ; @xj = 0 for all i; j). Definition 1.7. A dual to a vector bundle frame fv1(x); v2(x); : : : ; vn(x)g is a collection of n ∗ j i covector fields fv1(x); v2(x); : : : ; vn(x)g ⊂ Γ(T M) such that at each x 2 M, vi(v ) = δj. In most local computations, we shall use the coordinate frame. However, in some computa- tions, it is more convenient to use the orthonormal coordinate frame (for example, in the later parts of this note). Connections on the tangent bundle TM is particularly useful in studying the curvature of the manifold. For such connections r on TM, we define the Christoffel symbols which, by linearity and tensorial properties of connections, allow us to write the connection expression explicitly in local coordinates. n k Definition 1.8 (Christoffel symbols). In local coordinates fxigi=1, the Christoffel symbol Γij for a connection on TM is given by @ k @ r @ j = Γij k : @xi @x @x 2 @ k Remark 1.9 (Notation). We write @xi as @i so the above is simply r@i @j = Γij@k. Also, the Levi-Civita connection on TM induces a connection on T ∗M (which we also denote as r). By Remark 1.3, we can show: k k j k k j r @ dx = −Γijdx i.e. r@i dx = −Γijdx : @xi Remark 1.10. The Christoffel symbols for the Levi-Civita connection satisfy the following properties: 1. In local coordinates, the Christoffel symbols can be written explicitly in terms of the metric g as follows: 1 Γk = gkm (@ g + @ g − @ g ) : ij 2 j mi i mj m ij k k 2. It is symmetric i.e. Γij = Γji. 3. The following contraction identities are useful later: ij k 1 p ki i p g Γ = − @i( jgjg ) and Γ = @klog( jgj); ij pjgj ki where jgj = jdet(gij)j. 1.3 Second Covariant Derivative A connection DE on E together with a Levi-Civita connection on T ∗M induces a connection DT ∗M⊗E on T ∗M ⊗E. Therefore, we can compose the two connections to give a second covariant derivative: E T ∗M⊗E Γ(E) −−!D Γ(T ∗M ⊗ E) −−−−−−!D Γ(T ∗M ⊗ T ∗M ⊗ E): Definition 1.11 (Second covariant derivative). We define the second covariant derivative on a vector bundle E as the map D2 : Γ(E) ! Γ(T ∗M ⊗ T ∗M ⊗ E) given by 2 2 DXY σ = (D σ)(X; Y ) = (DX Dσ)(Y ) = DX (DY σ) − DrX Y σ; where r is the Levi-Civita connection. This expression is C1(M)-linear over X and Y . Unlike the second order derivative in Euclidean space, the second covariant derivatives generally do not commute. In other words, we get the following: 2 2 D σ(X; Y ) − D σ(Y; X) = DX (DY σ) − DY (DX σ) − D[X;Y ]σ: The expression above measures the failure for the second covariant derivative to commute. We can easily show that this quantity is in fact a tensor. This quantity is called the curvature of the connection, which is a central object of study in Riemannian geometry. 3 1.4 Curvatures on Vector Bundles Definition 1.12 (Curvature). The curvature of the connection D on the vector bundle E is the tensor R 2 Γ(T ∗M ⊗ T ∗M ⊗ E∗ ⊗ E) given by, for any X; Y 2 Γ(TM) and σ 2 Γ(E): R(X; Y )σ = D2σ(X; Y ) − D2σ(Y; X) = DX (DY σ) − DY (DX σ) − D[X;Y ]σ 2 Γ(E): (1) Definition 1.13 (Riemann curvature). When D is the Levi-Civita connection r of Riemannian metric g (and hence E is TM), the quantity R in Definition 1.12 is called the Riemann curvature. i n In local coordinates fx gi=1, we can write the Riemann curvature as: @ R = R ldxi ⊗ dxj ⊗ dxk ⊗ : ijk @xl We also have the Riemann curvature tensor, also denoted R, which is defined as: R(X; Y; Z; W ) = g(R(X; Y )Z; W ); for all W; X; Y; Z 2 Γ(TM).
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