We propose simultaneous confidence bands of the hyperbolic-type for the contrasts between several nonlinear (curvilinear) regression curves. The critical value of a confidence band is determined from the distribution of the maximum of a chi-square random process defined on the domain of explanatory variables. We use the volume-of-tube method to derive an upper tail probability formula of the maximum of a chi-square random process, which is asymptotically exact and sufficiently accurate in commonly used tail regions. Moreover, we prove that the formula obtained is equivalent to the expectation of the Euler-Poincare characteristic of the excursion set of the chi-square random process, and hence conservative. This result is therefore a generalization of Naiman's inequality for Gaussian random processes. As an illustrative example, growth curves of consomic mice are analyzed.