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. $$$$ ---------- **FOLLOW UP QUESTIONS** Following (almost) the terminology of Siu [6], a Riemannian manifold with nonpositive Hermitian sectional curvature has "strongly nonpositive curvature". He also introduces other notions of curvature such as "very strongly nonpositive" as follows. Consider the curvature operator $$ \begin{aligned} Q \colon \otimes^2 TM \times \otimes^2 TM \to \mathbb{R} \end{aligned} $$ such that $Q$ is defined for decomposable tensors by $Q(X\otimes Y, Z \otimes W) = R(X , Y, Z , W)$. By definition, *$N$ has very strongly nonpositive curvature* if $Q(\sigma, \sigma) \leqslant 0$ for all tensors $\sigma \in \otimes^2 (TM \otimes \mathbb{C})$ (not just decomposable ones). In other words, the curvature operator is positive semidefinite. **Question 2.** Is there an example showing that very strongly nonpositive curvature is strictly stronger than strongly nonpositive curvature? Finally, just to be thorough, there is a notion of (very) strongly negative curvature for Kähler manifolds, but it's not simply something like $Q(\sigma, \sigma) < 0$ for all nonzero $\sigma$, that is a bit too much to ask. Indeed, still denoting $Q$ its complex-linear extension, we have $Q(\sigma, \bar{\sigma}) = 0$ for any $\sigma$ of type $(2,0)$ or $(0,2)$, e.g. $X \otimes Y$ with $X, Y \in T^{1,0} M$. By definition, $N$ has very strongly negative curvature if $Q(\sigma, \bar{\sigma}) < 0$ for all nonzero tensors $\sigma \in T^{1,0} M \otimes T^{0,1} M$, and $N$ has strongly negative curvature if $Q(\sigma, \bar{\sigma}) < 0$ for all nonzero tensors $\sigma \in T^{1,0} M \otimes T^{0,1} M$ of length $\leqslant 2$, e.g. $\sigma = X \otimes \bar{Y} + Z \otimes {\bar{W}}$. It is clear that $$\text{very strongly negative} ~\Rightarrow~ \text{strongly negative} ~\Rightarrow~ \text{negative sectional curvature}$$ **Question 3.** Are there examples proving that the converse implications are false? Again, according to [9, Theorem 9.26], the answer is no for Kähler surfaces. Remark: Of course, there are similar notions of (very) strong nonnegative / positive curvature and one could ask the same questions. $$$$ ---------- [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.