Let $\mathcal{F}$ be a Riemannian foliation on a closed manifold M. Fix any Riemannian metric on the leaves that is smooth on M. Let V be any Riemannian vector bundle with a flat Riemannian $\mathcal{F}$ -partial connection. Let $p=\dim\mathcal{F}$ , $q=\operatorname{codim}\mathcal{F}$ and $k=\operatorname{rank} V$ . Let $(\Omega(\mathcal{F},V),d_\mathcal{F})$ be the leafwise de Rham complex with coefficients in V. Its reduced cohomology is denoted ${\mathcal H}(\mathcal{F},V)$ ; this is the maximal Hausdorff quotient of its cohomology, $H(\mathcal{F},V)/\overline0$ , using the topology on $H(\mathcal{F},V)$ induced by the $C^\infty$ topology on $\Omega(\mathcal{F},V)$ . Let $\delta_\mathcal{F}$ be the operator on $\Omega(\mathcal{F},V)$ defined by the de Rham coderivative on the leaves. Then $\Delta_\mathcal{F}:=d_\mathcal{F}\delta_\mathcal{F}+\delta_\mathcal{F} d_\mathcal{F}$ is the leafwise Laplacian on $\Omega(\mathcal{F},V)$ .

Since M is closed, the Hilbertian space $L^2\Omega(\mathcal{F},V)$ of $L^2$ leafwise differential forms with coefficients in V is well defined. The operators $d_\mathcal{F}$ , $\delta_\mathcal{F}$ and $\Delta_\mathcal{F}$ have closed extensions in $L^2\Omega(\mathcal{F},V)$ , denoted in the same way. Moreover, the leafwise heat equation defines the leafwise heat operator $e^{-t\Delta_\mathcal{F}}$ ( $t\gt 0)$ on $\Omega(\mathcal{F},V)$ [Reference RoeRoe87], which is bounded on $L^2\Omega(\mathcal{F},V)$ , where it is strongly convergent to a projection $\Pi:L^2\Omega(\mathcal{F},V)\to\mathrm{ker}\,\Delta_\mathcal{F}$ ; thus the notation $\Pi=e^{-\infty\Delta_\mathcal{F}}$ may be used.

A Hilbert-space scalar product on $L^2\Omega(\mathcal{F},V)$ can be defined in a standard way by using the product of the leafwise Riemannian density with any holonomy-invariant smooth non-vanishing transverse density (e.g., one induced by a bundle-like metric) and the Riemannian structure on V. Then $\delta_\mathcal{F}$ is the adjoint of $d_\mathcal{F}$ in $L^2\Omega(\mathcal{F},V)$ , and therefore $\Delta_\mathcal{F}$ is self-adjoint, $e^{-t\Delta_\mathcal{F}}$ on $L^2\Omega(\mathcal{F},V)$ can be given by the spectral theorem, and $\Pi$ is an orthogonal projection of $L^2\Omega(\mathcal{F},V)$ .

In [Reference Álvarez López and KordyukovÁLK01], Corollary C states that $\Pi$ also preserves $\Omega(\mathcal{F},V)$ , yielding a leafwise Hodge decomposition of $\Omega(\mathcal{F},V)$ (involving $\Delta_\mathcal{F}$ , $d_\mathcal{F}$ and $\delta_\mathcal{F}$ ), and a corresponding Hodge isomorphism ${\mathcal H}(\mathcal{F},V)\cong\mathrm{ker}\,\Delta_\mathcal{F}$ (induced by $\Pi$ ). However, S. Goette provided us with the counterexample explained in Example 1 below.

Let us recall our intended proof in [Reference Álvarez López and KordyukovÁLK01, Section 4] to explain our error and the solution. Since any metric on the leaves that is smooth on M can be extended to a bundle-like metric on M, Corollary C follows from Theorem B if V is a trivial Riemannian vector bundle with a trivial $\mathcal{F}$ -partial connection.

In the general case, we considered the principal O(k)-bundle of orthonormal frames of V, $\pi:F\to M$ , whose total space is a closed manifold. The flat metric $\mathcal{F}$ -partial connection on V can be understood as a completely integrable O(k)-invariant vector subbundle $H\subset TF$ so that $\pi_\ast:H_f\to T_{\pi(f)}\mathcal{F}$ is an isomorphism for every $f\in F$ . Let $\widehat{\mathcal{F}}$ be the corresponding foliation on F. As in Molino’s theory [Reference MolinoMol88], $\pi^\ast V$ is a trivial Riemannian vector bundle, and the lift of the flat $\mathcal{F}$ -partial connection on V is the trivial flat $\widehat{\mathcal{F}}$ -partial connection on $\pi^\ast V$ . Thus, if $\widehat{\mathcal{F}}$ were also a Riemannian foliation, Corollary C would hold for $(\Omega(\widehat{\mathcal{F}},\pi^\ast V), d_{\widehat{\mathcal{F}}})$ . Here, we could use any metric on the leaves of $\widehat{\mathcal{F}}$ that is smooth on F, and the corresponding operators $\delta_{\widehat{\mathcal{F}}}$ and $\Delta_{\widehat{\mathcal{F}}}$ .

Consider the injective homomorphism $\pi^\ast:(\Omega(\mathcal{F},V),d_\mathcal{F})\to(\Omega(\widehat{\mathcal{F}},\pi^\ast V),d_{\widehat{\mathcal{F}}})$ . Taking the lift of the given metric on the leaves of $\mathcal{F}$ to the leaves of $\widehat{\mathcal{F}}$ , the operators $\delta_{\widehat{\mathcal{F}}}$ and $\Delta_{\widehat{\mathcal{F}}}$ also correspond to $\delta_\mathcal{F}$ and $\Delta_\mathcal{F}$ by $\pi^*$ . Then Corollary C for $(\Omega(\mathcal{F},V), d_\mathcal{F})$ would follow from the case of $(\Omega(\widehat{\mathcal{F}},\pi^\ast V), d_{\widehat{\mathcal{F}}})$ .

We wrongly thought that $\widehat{\mathcal{F}}$ is always a Riemannian foliation, as in Molino’s theory, but this condition has to be assumed as a hypothesis. So the correct statement of Corollary C is the following.

Corollary C. With the above notation, assume that $\widehat{\mathcal{F}}$ is a Riemannian foliation. Then $\Pi$ defines a continuous operator on $\Omega(\mathcal{F},V)$ , we have the leafwise Hodge decomposition

$$\Omega(\mathcal{F},V)=\mathrm{ker}\,\Delta_\mathcal{F}\oplus\overline{\operatorname{im}\Delta_\mathcal{F}} =(\mathrm{ker}\, d_\mathcal{F}\cap\mathrm{ker}\,\delta_\mathcal{F})\oplus\overline{\operatorname{im} d_\mathcal{F}}\oplus\overline{\operatorname{im}\delta_\mathcal{F}}\;,$$

and $(t,\alpha)\mapsto e^{-t\Delta_\mathcal{F}}\alpha$ defines a continuous map $[0,\infty]\times\Omega(\mathcal{F},V)\to\Omega(\mathcal{F},V)$ . Thus ${\mathcal H}(\mathcal{F},V)$ can be canonically identified with $\mathrm{ker}\,\Delta_\mathcal{F}$ , and, if $\mathcal{F}$ is oriented, the leafwise Hodge star operator on $\mathrm{ker}\,\Delta_\mathcal{F}$ induces an isomorphism ${\mathcal H}^v(\mathcal{F},V)\cong{\mathcal H}^{p-v}(\mathcal{F},V^\ast)$ .

If V is the normal bundle $\nu=TM/T\mathcal{F}$ with the Bott $\mathcal{F}$ -partial connection and the Riemannian structure induced by any bundle-like metric, then F is the principal O(q)-bundle $\widehat M$ of orthonormal frames of $\nu$ . In this case, $\widehat{\mathcal{F}}$ is a Riemannian foliation by Molino’s theory. It easily follows that Corollary C is also true when V is a vector bundle associated to $\widehat M$ and any orthogonal representation of O(q), with the induced Riemannian structure and flat $\mathcal{F}$ -partial connection. For instance, this applies to the vector bundle $S^2(\nu)$ of symmetric bilinear forms on the fibers of $\nu$ , used in Section 5 of [Reference Álvarez López and KordyukovÁLK01]. Thus, the change in Corollary C does not affect the rest of the paper [Reference Álvarez López and KordyukovÁLK01].

Example 1 (S. Goette). Consider $\mathbb{S}^1\equiv\mathbb{R}/2\pi\mathbb{Z}$ , and let $[x]\in \mathbb{S}^1$ denote the class of every $x\in\mathbb{R}$ . Take the 2-torus $M = \mathbb{S}^1\times \mathbb{S}^1$ with the foliation $\mathcal{F}$ whose leaves are the fibers of the first factor projection to $\mathbb{S}^1$ . It is the quotient of the foliation $\widetilde{\mathcal{F}}$ on $\widetilde M :=\mathbb{R}^2$ whose leaves are the fibers of the first factor projection to $\mathbb{R}$ . Let V be the complex line bundle over M that is the quotient of the trivial complex line bundle $\widetilde V := \widetilde M\times\mathbb{C}$ over $\widetilde M$ by the action of $\mathbb{Z}^2$ defined by the transformations

\[T_{m,n}(x,y,z) = (x+2\pi m, y+2\pi n, e^{2\pi n \sin(x) i} z),\]

where $i = \sqrt{-1}$ and $m,n \in\mathbb{Z}$ . Consider also the trivial Hermitean structure on $\widetilde V$ . Then the trivial flat $\widetilde{\mathcal{F}}$ -partial connection $\widetilde{\nabla}$ on $\widetilde V$ is a Hermitean connection, and projects to a flat $\mathcal{F}$ -partial Hermitean connection $\nabla$ on V. Via the canonical identity $\mathbb{C} \equiv \mathbb{R}^2$ , $\widetilde V$ and V become oriented real vector bundles of rank 2, with trivial Riemannian structures so that $\widetilde{\nabla}$ and $\nabla$ are flat $\mathcal{F}$ -partial Riemannian connections. Then $\Omega^0(\mathcal{F}, V) = C^\infty(M; V)$ (the space of smooth sections of V), which can be considered as the subspace of $C^\infty(\widetilde M; \widetilde V) \equiv C^\infty(\widetilde M, \mathbb{C})$ consisting of the smooth functions $f: \widetilde M \to \mathbb{C}$ satisfying the equivariance condition

(1) \begin{equation}f(x+2\pi m, y+2\pi n) = e^{2\pi n \sin(x) i} f(x, y)\quad(m,n\in\mathbb{Z})\;.\end{equation}

Such a function f satisfies $d_{\widetilde{\mathcal{F}}}(f)=0$ if and only if it depends only on x; say $f = f(x)$ . Then Equation (1) means that

\[f(x) = e^{2\pi n \sin(x) i} f(x)\quad(n\in\mathbb{Z})\;.\]

If $\sin(x)$ is irrational, then the set $\{\,e^{2\pi m \sin(x) i}\mid m\in\mathbb{Z}\,\}$ is dense in the complex unit circle $\{\,z\in\mathbb{C}\mid|z|=1\,\}$ . The corresponding section $s \in C^\infty(M; V)$ is therefore invariant by multiplication by complex numbers of modulus 1 on the leaves $\{[x]\} \times \mathbb{S}^1$ when $\sin(x)$ is irrational, so s vanishes on those leaves, whose union is dense in M, and therefore s vanishes on M. This shows that $\mathrm{ker}\, d_\mathcal{F} = 0$ in $\Omega^0(\mathcal{F}, V)$ .

On the other hand, the function $f(x,y) = e^{y \sin(x) i}$ satisfies Equation (1) and is equal to 1 on $2\pi \mathbb{Z} \times \mathbb{R}$ , so it defines a section $s\in C^\infty(M;V)$ satisfying $\Delta_\mathcal{F} s = 0$ on the leaf $L_0 = \{ [0] \} \times \mathbb{S}^1$ . Then $s_t := e^{-t\Delta_\mathcal{F}}s$ remains independent of $t \gt 0$ on $L_0$ , and therefore the restriction to $2\pi \mathbb{Z} \times \mathbb{R}$ of its lift to $\widetilde M$ remains equal to f for all t. If the limit of $s_t$ as $t \to \infty$ were defined in $C^\infty(M;V)$ , it would be in $\mathrm{ker}\, d_\mathcal{F}$ and different from zero on $L_0$ , a contradiction.

In this case, the principal O(2)-bundle F of orthonormal frames of V over M is the quotient of the bundle $\widetilde F := \widetilde M\times \{\,z\in\mathbb{C}\mid|z|=1\,\}$ over $\widetilde M$ by the action of $\mathbb{Z}^2$ defined by the transformations $T_{m,n}$ ( $m,n \in\mathbb{Z}$ ). The foliation $\widehat{\mathcal{F}}$ on F is the quotient of the foliation $\widetilde{\mathcal{F}}$ on $\widetilde F$ whose leaves are the fibers of the first factor projection to $\mathbb{R}$ . The foliation $\widehat{\mathcal{F}}$ is not Riemannian because its leaf closures do not define a stratified structure of F (see Section 5.4 of [Reference MolinoMol88]).

We would like to take this opportunity to correct the following error on page 140 in [Reference Álvarez López and KordyukovÁLK01]. For $a \gt 1$ , the value of the integral $\int_0^{t/a}(1/{\sqrt{(t-s)s}})\,{ds}$ is $\pi/2+\arcsin(({2-a})/{a})$ instead of $\pi+\arctan(({2-a})/{a})$ . This correction gives the estimate needed there.