Direct numerical simulation is used to study the curvature of material surfaces in isotropic turbulence. The Navier–Stokes equation is solved by a 643pseudospectral code for constant‐density homogeneous isotropic turbulence, which is made statistically stationary by low‐wavenumber forcing. The Taylor‐scale Reynolds number is 39. An ensemble of 8192 infinitesimal material surface elements is tracked through the turbulence. For each element, a set of exact ordinary differential equations is integrated in time to determine, primarily, the two principal curvaturesk1andk2. Statistics are then deduced of the mean‐square curvatureM= (1)/(2) (k21+k22), and of the mean radius of curvatureR=(k21+k22)−1/2. Curvature statistics attain an essentially stationary state after about 15 Kolmogorov time scales. Then the area‐weighted expectation ofRis found to be 12&eegr;, where &eegr; is the Kolmogorov length scale. For moderate and small radii (less than 10&eegr;) the probability density function (pdf) ofRis approximately uniform, there being about 5% probability ofRbeing less than &eegr;. The uniformity of the pdf ofR, for smallR, implies that the expectation ofMis infinite. It is found that the surface elements with large curvatures are nearly cylindrical in shape (i.e., ‖k1‖≫‖k2‖ or ‖k2‖≫‖k1‖), consistent with the folding of the surface along nearly straight lines. Nevertheless the variance of the Gauss curvatureK=k1k2is infinite.