Bochner-Weitzenb\(\bf\ddot{o}\)ck Formula on Riemannian Manifolds
Let \((M,g)\) be an \(n\)-dimensional Riemannian manifold, and \(\nabla\) be the Levi-Civita connection. Recall that for a smooth function (or 0-form) \(f\in C^\infty(M)=\mathcal A^0(M)\), the Laplacian of smooth functions is defined as \[ \Delta_0(f)=\text{tr}\nabla^2(f)=\text{div}(\text{grad}(f)) \] The definition \(\Delta_0=\text{tr}\nabla^2\) can be generalized to arbitrary \(p\)-forms and is called the connection Laplacian. Suppose in addition \(M\) is oriented. Let \(d:\mathcal A^p(M)\to\mathcal A^{p+1}(M)\) be the differential operator. We can define the adjoint operator \(d^{\*}\) of \(d\) using the Hodge star operator. i.e. \(d^{\*}:\mathcal A^{p+1}(M)\to\mathcal A^p(M)\) is the operator satisfying \(\langle d\alpha,\beta\rangle=\langle\alpha,d^{\*}\beta\rangle=\int_Md\alpha\wedge {\*}\beta\) for \(\alpha\in\mathcal A^p(M)\) and \(\beta\in\mathcal A^{p+1}\). The Hodge Laplacian is defined by \[ \Delta=dd^*+d^*d \] By direct computation, we can see that \(\Delta=-\Delta_0\) on \(C^\infty(M)=\mathcal A^0(M)\). However, it is much more complicated in higher dimensions. The Bochner-Weitzenb\(\ddot{o}\)ck formula tells that, on \(\mathcal A^p(M)\) for \(p>0\), a term involving curvature tensor should be added. More precisely, we have \[ \Delta=-\Delta_0+\sum_{i,j}\omega^i\wedge i_{E_j}R_{E_iE_j} \] where \(E_1,\cdots,E_n\) is a local orthonormal frame and \(w^1,\cdots,\omega^n\) is the corresponding coframe. \(i_{E_j}\) is the interior multiplication and \(R_{E_iE_j}=\nabla_{[E_i,E_j]}-[\nabla_{E_i},\nabla_{E_j}]\) is the curvature tensor. For 1-forms the following equation is commonly presented in textbooks. \[ \frac{1}{2}\Delta_0|\omega|^2=|\nabla\omega|^2-\langle\Delta\omega,\omega\rangle+\text{Ric}(\omega_*,\omega_*) \] Using this formula one is able to prove the following Bochner's theorem
Let \(M\) be a compact, oriented Riemannian manifold, whose Ricci curvature tensor is nonnegative and positive at one point. Then \(H^1(M;\mathbb R)\)=0.
From Hodge theorem it suffices to prove that every harmonic 1-form is zero. Let \(\omega\) be a harmonic 1-form. Integrate equation (4) on both sides and note that on any compact manifold we have \(\int_M \Delta_0(f)dv=0\). We have \[ \nabla\omega=0 \text{ and } \text{Ric}(\omega_*,\omega_*)=0 \] For any vector field \(X\) we have \(X|\omega|^2=2\langle\nabla_X\omega,\omega\rangle=0\). Hence \(|\omega|\) is constant. If \(\omega\neq 0\) at one point then \(\omega\neq 0\) on \(M\). But \(\text{Ric}\) is positive at one point which means at this point \(\text{Ric}(\omega_\*,\omega_\*)>0\), which is a contradiction.
This 'standard' proof is not the same as Bochner's original proof. In fact, Bochner defined Laplacian in the following way. Since \(\nabla:\Gamma(\bigwedge^\*(M))\to\Gamma(\bigwedge^\*(M)\otimes T^\*M)\) is a linear map between inner product spaces, it admits an adjoint \(\nabla^\*\). The Bochner Laplacian is defined by \(\Delta_B=\nabla^\*\nabla\), which is also called rough Laplacian in literature (see Berger). It is easy to see that \(\Delta_B\) is nonnegative definite and \(\text{Ker}(\Delta_B)=\text{Ker}(\nabla)\) whose elements are parallel forms. By computation one verifies that \(\Delta_B=-\Delta_0\). Therefore, \[ \Delta=\Delta_B+\text{Curv}(R) \] If \(\text{Curv}(R)\) is nonnegative, then \(\text{Ker}(\Delta)=\text{Ker}(\Delta_B)\cap\text{Ker}(\text{Curv}(R))\). But parallel forms are completely determined at one point. Therefore, if \(\text{Curv}(R)\) is positive at one point, the harmonic forms will be identically zero.
Forman's Discretization of Ricci Curvature
Bochner's proof is carefully studied by Robin Forman so as to place a combinatorial analogy in discrete cases. In 2003, a series of combinatorial invariants for quasiconvex CW complexes were proposed, named 'curvature' by Forman. In the simplest case, suppose \(M\) is a simplicial complex. For each nonnegative integer \(p\), define the \(p\)th curvature function by \[ \mathcal F_p(\alpha)=\#\{(p+1)\text{-cofaces}\}+\#\{(p-1)\text{-faces}\}-\#\{\text{parallel neighbors}\} \] where parallel neighbors of \(\alpha\) are \(p\)-simplices sharing either a \((p+1)\)-coface or \((p-1)\)-face but not both. For \(p=1\), \(\mathcal F_1\) is called Ricci curvature by Forman, denoted by \(\text{Ric}\). Using this analogy Forman could prove several Bochner-type theorems. For example, one has the following
Let \(M\) be a connected weighted quasiconvex CW complex with nonnegative Ricci curvature. Suppose, in addition, there exists a vertex \(v\) such that \(\text{Ric}(e)>0\) for each coface \(e\) of \(v\). Then \(H_1(M,\mathbb R)=0\).
Forman's idea comes from a simple observation. Though the classical Bochner-Weitzenb\(\rm\ddot{o}\)ck formula looks rather abstruse, with many abstract operators on Riemannian manifolds involved, but from the simplest perspective, one has \[ \text{Laplacian}=\text{nonnegative definite operator}+\text{curvature} \]
For a simplicial complex \(M\) the discrete Laplacian is well defined. i.e. Let \(\partial_p:C_p(M;\mathbb R)\to C_{p-1}(M;\mathbb R)\) be the \(p\)th boundary operator. Place a metric on each chain vector space by declaring the simplices are orthogonal (It is not necessary that they are orthonormal. Note that when simplices are assigned weights the adjoint operator is no longer the transpose). The boundary operator admits an adjoint \(\partial_p^\*:C_{p-1}(M;\mathbb R)\to C_p(M;\mathbb R)\), such that \(\langle\partial_p\alpha,\beta\rangle_{p-1}=\langle\alpha,\partial_p^\*\beta\rangle_p\). The combinatorial Laplacian is defined by \[ \square_p=\partial_{p+1}\partial_{p+1}^*+\partial_p^*\partial_p:C_p(M;\mathbb R)\to C_p(M;\mathbb R) \] From equation (8) we know we want to decompose \(\square_p\) as a sum of a nonnegative definite matrix \(B_p\) and a curvature-type matrix \(F_p\). In fact we only need to construct \(B_p\), since we do not know any property of \(F_p\) so that we just put \(F_p=\square_p-B_p\).
The way Forman defined \(B_p\) is quite natural. Suppose \(A\) is a symmetric matrix. Then \[ \mathbb{B}(A)=\left\{\begin{array}{l}A_{ij},\text{ for }i\neq j\\ \sum_{j\neq i} |A_{ij}|,\text{ for }i=j\end{array}\right. \] is a diagonally dominant matrix, thus is nonnegative definite. \(\mathbb B(A)\) is called the Bochner matrix associated to \(A\). Thereafter \(\mathbb F(A)=A-\mathbb B(A)\) is called the curvature matrix associated to \(A\). From this perspective we see that 'curvature' measures how a symmetric matrix deviates from a diagonally dominant matrix. This decomposition is useful in our case because \(\mathbb B(\square)\) preserves the topological (or combinatorial) information of the simplicial complex \(M\). Specifically, if \(\alpha\) and \(\beta\) are parallel neighbors, then \(\square_{\alpha\beta}=\mathbb B(A)_{\alpha\beta}\neq 0\) by checking definition. Let \(B\) be a symmetric \(n\times n\) matrix. Define an equivalence relation on \(\{1,2,\cdots,n\}\) by requiring \(i\sim i\) and \(i\sim j\) if and only if there is a sequence \(i=k_0,k_1,\cdots,k_n=j\) such that \(B(k_l,k_{l+1})\neq 0\). Let \(\mathcal C(B)\) be the set of equivalence classes and \(\mathcal N(B)=|\mathcal C(B)|\). The following nontrivial property of diagonally dominant matrices is used by Forman to prove Bochner's theorem for 1-chains.
Let \(B\) be a diagonally dominant metrix, then
- \(\text{dim(ker)}(B)\le \mathcal N(B)\);
- Suppose \(v=(v_1,\cdots,v_n)\in\text{ker}(B)\), if \(B_{ij}\neq 0\), \(v_j=-sign(B_{ij})v_i\). i.e. the components in the same equivalence class are completely determined at one element.
It suffices to prove 2. Let \(c\in\mathcal C(B)\) be any class. Without loss of generality, assume \(v_i=\max_{j\in c}|v_j|\ge0\). Then we have \[ 0=(Bv)_i=\sum_{j}B_{ij}v_j=B_{ii}v_i+\sum_{j\in c,j\neq i}B_{ij}v_j\ge\sum_{j\in c,j\neq i}|B_{ij}|(v_i-|v_j|)\ge0 \] The equality holds if and only if \(v_j=0\) for all \(j\in c\), or, \(B_{ii}=\sum_{j\neq i}|B_{ij}|\) and \(v_j=-sign(B_{ij})v_i\) for all \(j\neq c\).
Recall that for graph Laplacian \(L=D-A\) the dimension of kernel is always equal to the connected components. This is because the constant vector \(\mathbf{1}\) provides a nontrivial element in the kernel if the graph is connected. However, one can easily construct an invertible matrix \(B\) with the diagonal equal to the sum of off-diagonal elements. Thus the first inequality can be strict.
Let us see how this property can be used in our case. Suppose \(\alpha\) and \(\beta\) are parallel \(p\)-neighbors, which implies \(\square_p(\alpha,\beta)\neq 0\) and thus \(\mathbb{B}(\square_p)(\alpha,\beta)\neq 0\). If there is a \(p\)-chain \(c=\sum c_\gamma\gamma\in\text{Ker}(\mathbb{B}(\square_p))\) with \(c_\alpha=0\), then \(c_\beta=0\). This continuation property will be important in the proof of Bochner-type theorems.
Now suppose \(\mathbb{F}(\square_p)\) is nonnegative definite, which is equivalent to say \(\mathbb{F}(\square_p)_{ii}\ge 0\). Being a sum of two nonnegative definite matrices, we have \(\text{Ker}(\square_p)=\text{Ker}(\mathbb{B}(\square_p))\cap\text{Ker}(\mathbb{F}(\square_p))\). In the simplest case, suppose \(\mathbb{F}(\square_p)\) is positive definite, then \(\text{Ker}(\square_p)=\{0\}\). For \(p=1\), we have the familiar statement: if \(M\) has positive Ricci curvature (at present call \(\mathbb{F}(\square_1)\) the Ricci curvature, which is positive if the diagonal elements are positive), then \(H_1(M;\mathbb R)\) is trivial.
More generally, assume \(M\) has nonnegative Ricci curvature and there is a vertex \(v\) such that \(\text{Ric}(e)>0\) for all \(e\succ v\). Let \(c=\sum c_e e\in\text{Ker}(\square_1)=\text{Ker}(\mathbb{B}(\square_1))\cap\text{Ker(Ric)}\). Since \(c\in\text{Ker(Ric)}\) we have \(c_e=0\) for all \(e\succ v\). The following lemma shows that \(c\) will be identically zero.
Suppose \(c=\sum c_e e\) is a 1-chain such that \(c\in\text{Ker}(\partial^\*)\cap\text{Ker}(\mathbb{B}(\square_1))\). In addition, there is a vertex \(v\) with \(c_e=0\) for all \(e\succ v\). Then \(c=0\).
Define \(D:\{\text{1-simplices}\}\to\mathbb{Z}_{\ge 0}\) as: (1). \(D(e)=0\) for \(e\succ v\); (2). Inductively, if \(D(e)\) is greater than \(k\) and there is a 1-simplex \(e_1\) such that \(e\cap e_1\neq\emptyset\) and \(D(e_1)=k\), set \(D(e)=k+1\).
From hypothesis \(c_e=0\) for \(D(e)=0\). Suppose \(c_e=0\) for all \(D(e)\le k\). Let \(e\) be a 1-simplex with \(D(e)=k+1\). Then there exists 1-simplex \(e_1\) such that \(D(e_1)=k\) and \(e\cap e_1\neq \emptyset\). If \(e\) and \(e_1\) are parallel, by the continuation property of \(\mathbb{B}(\square_1)\), \(c_e=0\). If \(e\) and \(e_1\) are not parallel, there is a 2-simplex \(f\) such that \(f\succ e\) and \(f\succ e_1\). Let the last edge of \(f\) be \(e_2\). Since \(D(e_1)=k\), either \(D(e)=k\) or \(D(e_2)=k\). By assumption \(D(e_2)=k\), thus \(c_{e_1}=c_{e_2}=0\). Then \[ \langle\partial f,c\rangle=\langle f,\partial^*c\rangle=0=\pm c_e\langle e,e\rangle \] Hence \(c=0\) by induction.
Now we give the explicit representation of \(\square_p\) and the curvature function \(\mathcal F_p\).
For each \((p+1)\)-simplex \(\beta\) with \(\partial\beta=\sum_{\alpha\prec\beta}\epsilon_{\alpha\beta}\alpha\) where \(\epsilon_{\alpha\beta}=\pm 1\) according to the orientation of \(\beta\). By definition, \(\langle\partial^\*\alpha,\beta\rangle=\langle\alpha,\partial\beta\rangle=\epsilon_{\alpha\beta}w_\alpha\) where we assume that \(\langle\alpha,\alpha\rangle=w_\alpha\). Thus \(\partial^\*\alpha=\sum_{\beta\succ\alpha}\epsilon_{\alpha\beta}\frac{w_\alpha}{w_\beta}\beta\). This gives \[ \square_p\alpha_1=\sum_{\alpha_2}[\sum_{\beta\succ\alpha_1,\alpha_2}\epsilon_{\alpha_1\beta}\epsilon_{\alpha_2\beta}\frac{w_{\alpha_1}}{w_\beta}+\sum_{\gamma\prec\alpha_1,\alpha_2}\epsilon_{\gamma\alpha_1}\epsilon_{\gamma\alpha_2}\frac{w_\gamma}{w_{\alpha_2}}]\alpha_2 \] We may work on orthonormal basis. Note that for different basis the definition of curvature will be different. Let \(\alpha^\*=\alpha/\sqrt{w_\alpha}\). Then \[ \langle\square_p\alpha_1^*,\alpha_2^*\rangle=\sum_{\beta\succ\alpha_1,\alpha_2}\epsilon_{\alpha_1\beta}\epsilon_{\alpha_2\beta}\frac{\sqrt{w_{\alpha_1}w_{\alpha_2}}}{w_\beta}+\sum_{\gamma\prec\alpha_1,\alpha_2}\epsilon_{\gamma\alpha_1}\epsilon_{\gamma\alpha_2}\frac{w_\gamma}{\sqrt{w_{\alpha_2}w_{\alpha_1}}} \] Denote by \(\square_p(\alpha_1^\*,\alpha_2^\*)\). Then under orthonormal basis, the curvature matrix is \[ \mathbb{F}(\square_p)(\alpha_1^*,\alpha_2^*)=\left\{\begin{array}{l}0,\text{ if }\alpha_1\neq\alpha_2\\ \square_p(\alpha_1^*,\alpha_2^*)-\sum_{\alpha^*\neq\alpha_2^*}|\square_p(\alpha^*,\alpha_1^*)|,\text{ if }\alpha_1=\alpha_2\end{array}\right. \] Define the \(p\)th curvature function by \(\mathcal{F}_p(\alpha)=w_\alpha\mathbb{F}(\square_p)(\alpha^\*,\alpha^\*)\). Thus \[ \begin{aligned}\mathcal{F}_p(\alpha)=w_\alpha(&\sum_{\beta\succ\alpha}\frac{w_\alpha}{w_\beta}+\sum_{\gamma\prec\alpha}\frac{w_\gamma}{w_\alpha}\\ &-\sum_{\eta\neq\alpha}|\sum_{\beta\succ\alpha,\eta}\epsilon_{\alpha\beta}\epsilon_{\eta\beta}\frac{\sqrt{w_{\alpha}w_{\eta}}}{w_\beta}+\sum_{\gamma\prec\alpha,\eta}\epsilon_{\gamma\alpha}\epsilon_{\gamma\eta}\frac{w_\gamma}{\sqrt{w_{\alpha}w_{\eta}}}|)\end{aligned} \] When \(p=1\) and all weights are 1, the formula simplifies as what is given in the beginning.