convolution without summation?

Hi, I'm new to deep learning. Here's the question about Convolutional Net: So a mn feature map can be mapped to a (m-k+1)(n-k+1) feature map by a kk filter. But I don't think it's a (complete) convolution. It seems that a complete convolution computing should further sum the (m-k+1)(n-k+1) features. However, the name and the mathematical function of convolution appeared in almost every materials make me confusing. I just want to know whether my understanding of Conv-Net is correct, that it's actually convolutions without summations.