Return to Answer

Yes, there is a Hodge decomposition for elliptic complexes on compact oriented Riemannian manifolds.

Let $(M,g)$ be a compact oriented Riemannian manifold and let $$ 0 \to \Gamma(E_0) \to \Gamma(E_1) \to \dots \to \Gamma(E_m) \to 0 $$ be an elliptic differential complex with the arrows given by the differential operators $D_0,\dotsc,D_{m-1}$, where each $E_i$ is equipped with a metric and a compatible connection. Thus we have formal adjoints and can form the Laplacians: $$ \Delta_i := D_i^* \circ D_i + D_{i-1} \circ D_{i-1}^*: \Gamma(E_i) \to \Gamma(E_i). $$ Then we get the Hodge decomposition $$ L^2(E_i) = \ker(\Delta_i) \oplus \operatorname{ran}(D_{i-1}) \oplus \operatorname{ran}(D_i^*). $$

I learnt this from our own Liviu Nicolaescu's Lectures on the Geometry of Manifolds. More precisely, this is Chapter 10.4.3 in the notes (my version is from 9. Sep 2018).

Yes, there is a Hodge decomposition for elliptic complexes on compact oriented Riemannian manifolds.

$L^2$-version. Let $(M,g)$ be a compact oriented Riemannian manifold and let $$ 0 \to \Gamma(E_0) \to \Gamma(E_1) \to \dots \to \Gamma(E_m) \to 0 $$ be an elliptic differential complex with the arrows given by the differential operators $D_0,\dotsc,D_{m-1}$, where each $E_i$ is equipped with a metric and a compatible connection. Thus we have formal adjoints and can form the Laplacians: $$ \Delta_i := D_i^* \circ D_i + D_{i-1} \circ D_{i-1}^*: \Gamma(E_i) \to \Gamma(E_i). $$ Then we get the Hodge decomposition $$ L^2(E_i) = \ker(\Delta_i) \oplus \operatorname{ran}(D_{i-1}) \oplus \operatorname{ran}(D_i^*). $$

I learnt this from our own Liviu Nicolaescu's Lectures on the Geometry of Manifolds. More precisely, this is Chapter 10.4.3 in the notes (my version is from 9. Sep 2018).

EDIT: As per OP's comment, I am adding the smooth version I know of.

$C^{\infty}$-version. Consider the total vector bundle $E := \bigoplus_{i=0}^{m} E_i$ and the Laplacian $\Delta = D \circ D^* + D^* \circ D$, i.e. $\Delta$ is just a tuple of Laplacians $\Delta_i: \Gamma(E_i) \to \Gamma(E_i)$. The following statements are equivalent:

(1) $\Delta$ is Fredholm and $\Gamma(E) = \ker(\Delta) + \operatorname{ran}(\Delta)$ (note, this is just a sum, not a direct sum);

(2) The complex itself is Fredholm and satisfies the Hodge decomposition: $$ \Gamma(E) = \ker(\Delta) \oplus \operatorname{ran}(D) \oplus \operatorname{ran}(D^*) $$ This can be found in Chapter 1 of van den Ban and Crainic's lectures on Analysis on Manifolds, more precisely exercise 1.3.15, if you don't mind following the exercises and working it out from the preceding theory, but I don't know a more self-contained source.

Yes, there is Hodge decomposition for elliptic complexes on compact oriented Riemannian manifolds.

Let $(M,g)$ be a compact oriented Riemannian manifold and let $$ 0 \to \Gamma(E_0) \to \Gamma(E_1) \to \dots \to \Gamma(E_m) \to 0 $$ be an elliptic differential complex with the arrows given by the differential operators $D_0,\dots,D_{m-1}$, where each $E_i$ is equipped with a metric and a compatible connection. Thus we have formal adjoints and can form the Laplacians: $$ \Delta_i := D_i^* \circ D_i + D_{i-1} \circ D_{i-1}^*: \Gamma(E_i) \to \Gamma(E_i) $$ Then we get the Hodge decomposition $$ L^2(E_i) = \ker(\Delta_i) \oplus \operatorname{ran}(D_{i-1}) \oplus \operatorname{ran}(D_i^*) $$

I learnt this from our own Liviu Nicolaescu's Lectures on the Geometry of Manifolds. More precisely, this is Chapter 10.4.3 in the notes (my version is from 9. Sep 2018).

Yes, there is a Hodge decomposition for elliptic complexes on compact oriented Riemannian manifolds.

Let $(M,g)$ be a compact oriented Riemannian manifold and let $$ 0 \to \Gamma(E_0) \to \Gamma(E_1) \to \dots \to \Gamma(E_m) \to 0 $$ be an elliptic differential complex with the arrows given by the differential operators $D_0,\dotsc,D_{m-1}$, where each $E_i$ is equipped with a metric and a compatible connection. Thus we have formal adjoints and can form the Laplacians: $$ \Delta_i := D_i^* \circ D_i + D_{i-1} \circ D_{i-1}^*: \Gamma(E_i) \to \Gamma(E_i). $$ Then we get the Hodge decomposition $$ L^2(E_i) = \ker(\Delta_i) \oplus \operatorname{ran}(D_{i-1}) \oplus \operatorname{ran}(D_i^*). $$

I learnt this from our own Liviu Nicolaescu's Lectures on the Geometry of Manifolds. More precisely, this is Chapter 10.4.3 in the notes (my version is from 9. Sep 2018).

Yes, there is Hodge decomposition for elliptic complexes on compact oriented Riemannian manifolds.

Let $(M,g)$ be a compact oriented Riemannian manifold and let $$ 0 \to \Gamma(E_0) \to \Gamma(E_1) \to \dots \to \Gamma(E_m) \to 0 $$ be an elliptic differential complex with the arrows given by the differential operators $D_0,\dots,D_{m-1}$, where each $E_i$ is equipped with a metric and compatible connection. Thus we have formal adjoints and can form the Laplacians: $$ \Delta_i := D_i^* \circ D_i + D_{i-1} \circ D_{i-1}^*: \Gamma(E_i) \to \Gamma(E_i) $$ Then we get the Hodge decomposition $$ L^2(E_i) = \ker(\Delta_i) \oplus \operatorname{im}(D_{i-1}) \oplus \operatorname{im}(D_i^*) $$

Yes, there is Hodge decomposition for elliptic complexes on compact oriented Riemannian manifolds.

Let $(M,g)$ be a compact oriented Riemannian manifold and let $$ 0 \to \Gamma(E_0) \to \Gamma(E_1) \to \dots \to \Gamma(E_m) \to 0 $$ be an elliptic differential complex with the arrows given by the differential operators $D_0,\dots,D_{m-1}$, where each $E_i$ is equipped with a metric and a compatible connection. Thus we have formal adjoints and can form the Laplacians: $$ \Delta_i := D_i^* \circ D_i + D_{i-1} \circ D_{i-1}^*: \Gamma(E_i) \to \Gamma(E_i) $$ Then we get the Hodge decomposition $$ L^2(E_i) = \ker(\Delta_i) \oplus \operatorname{ran}(D_{i-1}) \oplus \operatorname{ran}(D_i^*) $$

I learnt this from our own Liviu Nicolaescu's Lectures on the Geometry of Manifolds. More precisely, this is Chapter 10.4.3 in the notes (my version is from 9. Sep 2018).