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Curvature conservation is a common feature for the physical properties with topological constraints, from the macroscopic cosmic string loops to the microscopic quantized vortex rings in the superfluid. Interestingly, ferroelectric domain-wall curvature is important to determine its conduction, which is the key parameter for designing domain-wall nanoelectronic devices. Here, we demonstrate the curvature conservation of the charged domain walls confined in ferroelectric topological structures with specific domain symmetry by combining piezoelectric force microscopy, conducting atomic force microscopy, and phase-field simulations. Significantly, the intrinsic conductivity of the charged domain walls exhibits an inverse functional relationship with the local domain-wall curvature in the ferroelectric BiFeO 3 nano-islands, which can also be precisely and continuously modulated by applying an electric field. This work provides us with an insightful understanding of the conduction mechanism of charged domain walls confined in ferroelectric topological structures with a high degree of spatial symmetry, which paves the way for the application of charged domain walls as flexible conducting channels with continuously changing resistance states to be used for ferroelectric domain-wall memristors. [Display omitted] [ABSTRACT FROM AUTHOR]