We address a problem of optimization on product of embedded submanifolds of convolution kernels (PEMs) in convolutional neural networks (CNNs). First, we explore metric and curvature properties of PEMs in terms of component manifolds. Next, we propose a SGD method, called C-SGD, by generalizing the SGD methods employed on kernel submanifolds for optimization on product of different collections of kernel submanifolds. Then, we analyze convergence properties of the C-SGD considering sectional curvature properties of PEMs. In the theoretical results, we expound the constraints that can be used to compute adaptive step sizes of the C-SGD in order to assure the asymptotic convergence.