Van der Waerden-Bortolotti Curvature Operator

• The Van der Waerden-Bortolotti curvature operator is defined as a self-adjoint extension of the Riemann curvature tensor acting on symmetric (0,2)-tensors on Riemannian manifolds.
• It provides a framework for deriving refined curvature pinching and topological rigidity results through eigenvalue analysis and Bochner-Weitzenböck techniques.
• The operator is central to proving Betti number vanishing theorems and establishing spectral thresholds that classify manifolds under weakened curvature conditions.

The Van der Waerden-Bortolotti curvature operator, also known as the curvature operator of the second kind, extends the classical action of the Riemann curvature tensor to the space of symmetric (0,2)-tensors on Riemannian manifolds. Unlike the standard curvature operator on 2-forms, the Van der Waerden-Bortolotti operator plays a fundamental role in refined curvature pinching results, topological rigidity theorems, and Betti number vanishing via Bochner-Weitzenböck techniques. The algebraic and analytical properties of this operator, including its eigenvalue structure and positivity conditions, are central to recent developments in global differential geometry, especially those concerning topological characterizations of manifolds under weakened curvature hypotheses (Nienhaus et al., 2022).

1. Definition and Algebraic Structure

Given an $n$-dimensional Riemannian manifold $(M,g)$ with Riemann curvature tensor $R_{ijkl}$, the curvature tensor is typically realized as an operator on $\Lambda^2T_pM$ via

$$R^{(1)} : \Lambda^2T_pM \to \Lambda^2T_pM,\quad (R^{(1)}(W))_{ij} = R_{ij}{}^{kl}W_{kl}.$$

Following Van der Waerden and Bortolotti, and later Bourguignon–Karcher, an alternative construction defines the curvature operator on the vector space of symmetric $(0,2)$-tensors, $S^2T_p^*M$:

$R^{(2)} : S^2T_p^*M \longrightarrow S^2T_p^*M,$

with the action

$$(R^{(2)}h)_{ij} = g^{k\ell}R_{i\,k\,\ell\,j}h_{k\ell},$$

for $h \in S^2T_p^*M$. In an orthonormal frame $\{e_a\}$, this can also be written as

$$R^{(2)}(h) = \sum_{a<b}(e^a\odot e^b)\langle R(e^a\wedge e^b), h \rangle.$$

$R^{(2)}$ is self-adjoint with respect to the inner product

$$\langle h, k \rangle = g^{ij}g^{k\ell}h_{ik}k_{j\ell},$$

ensuring real eigenvalues. The dimension of $S^2T_p^*M$ is $\frac{1}{2}n(n+1)$, but for curvature operator eigenvalue discussions, the ordering uses $N = \frac{1}{2}(n-1)(n+2)$ eigenvalues $\mu_1 \leq \mu_2 \leq \cdots \leq \mu_N$.

2. Positivity, Nonnegativity, and Spectral Thresholds

Positivity conditions for $R^{(2)}$ are expressed through $k$-nonnegativity and $k$-positivity:

• $R^{(2)}$ is $k$-nonnegative if

$$\sum_{a=1}^{[k]} \mu_a + (k - [k])\mu_{[k]+1} \geq 0$$

for real $k \geq 1$.

• $k$-positivity demands strict positivity of the same weighted sum.
• When $k=1$, this reduces to standard nonnegativity of the lowest eigenvalue; for $k=2$, $\mu_1 + \mu_2 > 0$. More generally, "C-positivity" refers to $k$-positivity for real $C>0$. A significant family of thresholds arises in Bochner-type vanishing theorems, parameterized by $C_p(n)$ for each $p$:

$$C_p(n) = \frac{3\big(2n^2p - p^2n - 2np + 2n^2 + 2n - 4p\big)}{n(n+2)p(n-p)}.$$

$R^{(2)}$ is $C_p(n)$–positive if it is $k$–positive for $k = C_p(n)$. As $p$ increases, $C_p(n)$ decreases, so the associated curvature positivity conditions weaken.

3. Bochner-Weitzenböck Techniques and Harmonic Forms

For harmonic $p$-forms, the Weitzenböck formula is

$$\Delta\omega = \nabla^*\nabla\omega + \mathrm{Ric}^{(L)}(\omega),$$

with the Bochner identity

$$\frac{1}{2}\Delta|\omega|^2 = |\nabla\omega|^2 + \langle\mathrm{Ric}^{(L)}(\omega), \omega\rangle.$$

The induced curvature term decomposes as follows [(Nienhaus et al., 2022), Prop. 2.1]:

$$\langle\mathrm{Ric}^{(L)}(\omega),\omega\rangle = \langle R^{(2)}(\omega_{[2]}), \omega_{[2]}\rangle + \frac{3p(n-2p)}{n}R_{ijk\ell}\omega_{ji2\ldots p}\omega_{k i2\ldots p} + \frac{p^2}{n^2}\mathrm{scal}|\omega|^2$$

where $\omega_{[2]}$ denotes the component in the "S^2-bundle" and the remaining terms involve Ricci and scalar curvatures. The "weight principle" (Theorem 3.6 in (Nienhaus et al., 2022)) shows that sufficient $k$-nonnegativity of $R^{(2)}$ ensures the nonnegativity of each summand, forcing any harmonic form to be parallel, and, if $\mathrm{scal}>0$ at some point, to vanish identically.

4. Topological Implications: Betti Number Vanishing and Rigidity

The principal geometric consequences of $R^{(2)}$-positivity or nonnegativity are encapsulated by a hierarchy of Bochner-type vanishing and rigidity theorems (Nienhaus et al., 2022):

• Theorem A: If $R^{(2)}$ is $n(n-1)$–nonnegative, every harmonic $p$–form is parallel. Unless $M$ is flat, all harmonic forms vanish; $M$ is a rational homology sphere.
• Corollary (Li [Li22] + Theorem A): If $n\geq4$ and $R^{(2)}$ is $3$–nonnegative, $M$ is either flat or diffeomorphic to a spherical space form.
• Theorem B (Einstein Case): For $N=\frac{3n(n+2)}{2(n+4)}$,
  • If $R^{(2)}$ is $N$–positive, $M$ is a rational homology sphere.
  • If $R^{(2)}$ is $N'$–nonnegative for $N'<N$, $M$ is either flat or a rational homology sphere.
  • If $R^{(2)}$ is $N$–nonnegative, every harmonic form is parallel.
• Theorem C (Betti Number Vanishing): For $C_p(n)$ as above and $0
• If $R^{(2)}$ is $C_p(n)$–positive, $b_p(M) = 0$.
• If $R^{(2)}$ is $C'$–nonnegative for $C'<C_p(n)$, $b_p(M)=0$ or $M$ is flat.
• If $R^{(2)}$ is $C_p(n)$–nonnegative, every harmonic $p$–form is parallel, so aside from the flat case, $b_p=0$ for $p\neq0,n$.

These results extend classical Bochner theory: for large enough $p$, the curvature condition for Betti number vanishing becomes strictly weaker.

5. Model Spaces and Explicit Examples

Four canonical examples illustrate the spectrum and topological implications of $R^{(2)}$ (Nienhaus et al., 2022):

Manifold	$R^{(2)}$ Spectrum	Positivity/Nonnegativity
Round sphere $S^n$	$\mu_a=1$ (all $a$)	$k$–positive for all $k>0$
Flat $\mathbb{R}^n$	$0$	$k$–nonnegative for all $k\geq1$
Product $S^1 \times S^{n-1}$	$-2$ (mult. 1), $0$ (mult. $n-1$), $+1$ (mult. $N-n$)	$(n+1)$–positive, not $n$–nonnegative
$SU(3)/SO(3)$ ($n=5$)	$-3/2$ (mult. 5), $+2$ (mult. 9)	$9$–positive, not $8$–nonnegative

A notable phenomenon is that even when $R^{(2)}$ is $C$–positive for large $C$, Ricci curvature can be negative, as with $S^1\times S^{n-1}$.

6. Quantitative and Diameter-Dependent Results

Beyond pointwise curvature positivity, diameter-dependent upper bounds for Betti numbers are derived under average lower bounds for $R^{(2)}$ and finite diameter constraints. Classical techniques of Gallot and Li are adapted to the context of $R^{(2)}$ (see Theorem D in (Nienhaus et al., 2022)). A plausible implication is that the interplay between geometry and topology controlled by $R^{(2)}$ is robust under weaker, averaged conditions, thus broadening the landscape of vanishing and rigidity results even when strict pointwise thresholds fail.

7. Historical Context and Further Directions

The construction of $R^{(2)}$ originates with Van der Waerden and Bortolotti, with subsequent refinements by Bourguignon–Karcher. The modern utility of $R^{(2)}$ is in its finer sensitivity to symmetric tensors, enabling topological and analytic conclusions inaccessible through the classical $R^{(1)}$. The comprehensive treatment by Nienhaus, Petersen, and Wink (Nienhaus et al., 2022) establishes the central role of $R^{(2)}$ in understanding the relationship between curvature positivity, Betti number vanishing, and manifold rigidity, setting the stage for further investigations of geometric-topological interdependence in Riemannian geometry.