I'm reading "Elements of Noncommutative Geometry" by Garcia-Bondía and have troubles to understand the connection Laplacians. The following was written
First scalar Laplacian were considered, i.e. acting on section of the trivial line bundle $L=M\times\mathbb{C}\to M$. A natural generalisation of the Laplacian would be given by $-g^{ij}\nabla^g_{\partial_i}\nabla^g_{\partial_j}$.
To understand this formula, it was mentioned that a connection on the trivial line bundle reduces to $d:\Gamma^\infty(L)\to\Omega^1(M,L)=\Gamma^\infty(T^*M\otimes L)$. For thw Levi-Civita connection $\nabla^g$ holds: $$\nabla^g:\Gamma^\infty(T^*M)=\Gamma^\infty(T^*M\otimes L)\to\Omega^1(M,T^*M\otimes L)=\Gamma^\infty(T^*M\otimes T^*M\otimes L)$$
Contracting with the metric $g^{-1}$ on $T^*M$ gives a $C^\infty(M)$-linear map $\mathrm{Tr}_g:\Gamma(T^*M\otimes T^*M\otimes L)\to\Gamma(L)$. The composition of the 3 operators yields an operator on the sections of $L$ $$\Delta:=-\mathrm{Tr}_g\circ\nabla^g\circ d$$
Defining the non-scalar connection Laplacian, consider the line bundle $L\to M$. A general connection on $L$ has the form $\nabla^L=d+\alpha$, where the connection 1-form $\alpha=iA_jdx^j$ has only one matrix component. Define the connection Laplacian as $$\Delta^L=-\mathrm{Tr}_g\circ\tilde{\nabla^L}\circ\nabla^L,$$ where $\tilde{\nabla^L}$ is the tensor product of $\nabla^L$ and $\nabla^g$.
I do not see the connection between $-g^{ij}\nabla^g_{\partial_i}\nabla^g_{\partial_j}$ and the operator $\Delta$. In general one can write a connection $\nabla=d+\alpha$ with $\alpha$ the connection 1-form. Could you please explain why the connection 1-form vanishes for the scalar case and has only one matrix element for the non-scalar case.
Thanks for your help.
The most important thing to note is that the trivial line $M\times \mathbb{C}$ has a natural connection on it given by the exterior derivative, $d: \Gamma(M\times \mathbb{C})=C^\infty(M,\mathbb{C})\to \Gamma(T^*M\otimes \mathbb{C})=\Omega^1(M,\mathbb{C})$, but in general there are many choices that can be made. The definition of a connection is a map $\nabla: \Gamma(L)\to \Gamma(L\otimes T^*M)$ which is $\mathbb{R}$ linear and obeys the Leibniz rule $\nabla(f\sigma)=f\nabla\sigma+\sigma\otimes df$ for all $\sigma\in \Gamma(L)$ and $f\in C^\infty(M)$. We can see that given $\nabla$ a connection on $L$, $\nabla+A$ is a connection on $L$ if $A\in \Gamma(\mathrm{End}(L)\otimes T^*M)\cong\Omega^1(M,\mathbb{C})$ since $(\nabla+A)(f\sigma)=f\nabla\sigma+\sigma\otimes df+f\sigma\otimes A=f(\nabla\sigma+\sigma\otimes A)+\sigma\otimes df=f(\nabla+A)\sigma+\sigma\otimes df$. In fact, the difference of any two connections $\nabla-\nabla'$ is an element of $\Omega^1(M,\mathbb{C})$ (Check this!).
Since any line bundle is locally of the form $U\times \mathbb{C}$ we can say that locally any connection is given $d+A$ where $A$ is an element of $\Omega^1(U,\mathbb{C})$. Of course, on a general line bundle there is no global notion of $d$ so this statement only makes sense locally.
Working on $M\times \mathbb{C}$ we have the god given connection from the exterior derivative, so this is the one we begin with in order to define a Laplacian. In coordinates we have $d(\sigma)=\frac{\partial \sigma}{\partial x^i}dx^i$ (this is actually the same as $\nabla_g\sigma$ for the Levi-Civita connection, by convention.) Then $\nabla_g(d\sigma)=\left(\frac{\partial^2\sigma}{\partial x^j\partial x^i} -\Gamma^l_{ij}\frac{\partial\sigma}{\partial x^l}\right)dx^i\otimes dx^j.$ Applying the metric gives the endomorphism $g^{ik}\left(\frac{\partial^2\sigma}{\partial x^j\partial x^i} -\Gamma^l_{ij}\frac{\partial\sigma}{\partial x^l}\right)\partial_{x^k}\otimes dx^j$. The trace of this gives $g^{ij}\left(\frac{\partial^2\sigma}{\partial x^j\partial x^i}-\Gamma^l_{ij} \frac{\partial \sigma}{\partial x^l}\right)$, which of course is $g^{ij}\nabla_i\nabla_j\sigma$.
Since $d$ is a connection for the trivial line, and $\nabla_g$, the Levi-Civita conection can be thought of as a connection on $T^*M\otimes \mathbb{C}$, a natural generalization of $\Delta_g=-\mathrm{tr}_g\circ \nabla_g\circ d$ to a line bundle is to replace the terms $\nabla_g$ and $d$ by $\tilde{\nabla}^L$ and $\nabla^L$ respectively, where $\nabla^L$ is a connection on $L$ and $\tilde{\nabla}^L$ is the connection on $T^*M\otimes L$ induced by $\nabla^L$ and $\nabla_g$ the Levi-Civita connection, yielding $$\Delta=-\mathrm{tr}_g \circ \tilde{\nabla}^L\circ \nabla^L$$