The information geometry of the zero-mean t-distribution with Kronecker-product structured covariance matrix is derived. In particular, we obtain the Fisher information metric which shows that this geometry is identifiable to a product manifold of S++p (positive definite symmetric matrices) and sS++p (positive definite symmetric matrices with unit determinant). From this result, we obtain the geodesics and the Riemannian gradient. Finally, this geometry makes it possible to propose an on-line covariance matrix estimation algorithm well adapted to large datasets. Numerical experiments show that optimal results are obtained for a reasonable number of data.