Capturing and preserving physical properties, e.g., system energy, stability and passivity, using data-driven methods is currently a highly-researched topic in surrogate modeling. To ensure that the desired physical properties are retained, structure-preserving projection techniques are used in the field in model reduction (MOR). In this talk, we present structure-preserving MOR with nonlinear projections, which are needed for problems with slowly decaying Kolmogorov-$n$-widths. In particular, we focus on the reduction of the class of port-Hamiltonian systems. Based on a differential geometric framework, we derive novel MOR methods, ensuring that the pH structure in the reduced-order model (ROM) is preserved. Further, we present numerical results, which show that the new presented methods preserve the interconnectedness property within the port-Hamiltonian ROMs.