Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair.
Consider the complexified tangent space $TM \otimes \mathbb{C}$ and the complex-linear extension of $R$, which we still denote $R$. By definition, $N$ has nonpositive *Hermitian sectional curvature* if $R(X, Y, \bar{X}, \bar{Y}) \leqslant 0$ for all $X, Y \in TM \otimes \mathbb{C}$.
Obviously, nonpositive Hermitian sectional curvature is stronger than nonpositive sectional curvature.
**QUESTION.** Is nonpositive Hermitian curvature *strictly* stronger than nonpositive curvature?
In other words, are there examples of Riemannian manifolds with nonpositive sectional curvature, but not nonpositive Hermitian sectional curvature?
I expect the answer easily yes, in fact it is claimed in e.g. [1] or [8], but I couldn't find an example in the relevant literature, e.g. [1][2][3][4][5][6][7][8][9].
NB: Yau-Zheng [8] showed that the answer is no for manifolds with negative $\delta$-pinched sectional curvature with $\delta \geqslant 1/4$. According to [9, Theorem 9.26], the answer is no for Kähler surfaces.
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**FOLLOW UP QUESTIONS**
Following (almost) the terminology of Siu [6], a Riemannian manifold with nonpositive Hermitian sectional curvature has "strongly nonpositive curvature". He also introduces other notions of curvature such as "very strongly nonpositive" as follows. Consider the curvature operator
$$
\begin{aligned}
Q \colon \Lambda^2 TM \times \Lambda^2 TM \to \mathbb{R}
\end{aligned}
$$
such that $Q$ is defined for decomposable tensors by $Q(X\wedge Y, Z \wedge W) = R(X , Y, Z , W)$. We still denote $Q$ its complex-linear extension to complexified vectors. By definition, *$N$ has very strongly nonpositive curvature* if $Q(\sigma, \bar{\sigma}) \leqslant 0$ for all tensors $\sigma \in \Lambda^2 TM \otimes \mathbb{C}$ (not just decomposable ones).
**Question 2.** Is there an example showing that very strongly nonpositive curvature is strictly stronger than strongly nonpositive curvature?
**Question 3.** What about the condition that $Q(\sigma, \sigma) \leqslant 0$ for all $\sigma \in \Lambda^2 TM$ (no complexification)? Is it stronger than nonpositive curvature?
Finally, just to be thorough, there is a notion of (very) strongly negative curvature, but it's not simply something like $Q(\sigma, \bar{\sigma}) < 0$ for all nonzero sigma, because that is too much to ask. Assume now that $N$ is a Kähler manifold. Then $Q(\sigma, \bar{\sigma}) = 0$ for any $\sigma$ of type $(2,0)$ or $(0,2)$, e.g. $X \wedge Y$ with $X, Y \in T^{1,0} M$. By definition, $N$ has very strongly negative curvature if $Q(\sigma, \bar{\sigma}) < 0$ for all nonzero tensors $\sigma$ of type $(1,1)$, and $N$ has strongly negative curvature if $Q(\sigma, \bar{\sigma}) < 0$ for all nonzero tensors $\sigma$ of type $(1,1)$ and of length $\leqslant 2$, e.g. $\sigma = X \wedge \bar{Y} + Z \wedge{\bar{W}}$.
It is clear that
$$\text{very strongly negative} ~\Rightarrow~ \text{strongly negative} ~\Rightarrow~ \text{negative sectional curvature}$$
**Question 4.** Are there examples proving that the converse implications are false?
Again, according to [9, Theorem 9.26], the answer is no for Kähler surfaces.
Remark: Of course, there are similar notions of (very) strong nonnegative / positive curvature and one could ask the same questions.
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[1] J. Amorós, M. Burger, K. Corlette, D. Kotschick, and D. Toledo. Fundamental groups of compact Kähler manifolds. 1996.
[2] Eells and Lemaire. Two reports on harmonic maps. 1995
[3] Jost and Yau. Harmonic mappings and Kähler manifolds. 1983.
[4] Mostow and Siu. A compact Kähler surface of negative curvature not coveredby the ball. 1980.
[5] Ohnita and Udagawa. Stability, complex-analyticity and constancy of pluriharmonic maps from compact Kaehler manifolds. 1990.
[6] Siu. The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. 1980.
[7] Xin. Geometry of harmonic maps. 1996
[8] Yau and Zheng. Negatively $\frac14$-pinched Riemannian metric on a compact Kähler manifold.
[9] F. Zheng, Complex differential geometry, 2000.