In the page 79 of Griffiths' and Harris' Principles of Algebraic Geometry it says that the curvatures satisfy $$ \Theta_S\leq \Theta_E|_S,\quad \Theta_Q\geq \Theta_E|_Q $$ for $E$ a Hermitian bundle, $S\subset E$ a holomorphic subbundle and $Q$ the quotient bundle.
If $M\subset\mathbb{C}^n$ is a complex submanifold, for example, a Riemann surface, whose metric is induced from the Euclidean metric in $\mathbb{C}^n$, then $\Theta_M\leq \Theta_{\mathbb{C}^n}|_M = 0$. If $\Phi$ is the associated $(1,1)$-form then $\sqrt{-1}\Theta=K\Phi$ where $K$ is the Gaussian curvature (page 77 of Griffiths-Harris). Since $\Phi$ is positive, we must have $K\leq 0$.
My question is, if we take $M$ to be the Riemann sphere, then it has positive Gaussian curvature, which contradicts $K\leq 0$, so what is the problem here?