  In this paper, the first Chen inequality is proved for general warped product
submanifolds in Riemannian space forms, this inequality involves intrinsic
invariants ($\delta$-invariant and sectional curvature) controlled by an
extrinsic one (the mean curvature vector), which provides an answer for Problem
1. As a geometric application, this inequality is applied to derive a necessary
condition for the immersed submanifold to be minimal in Riemannian space forms,
which presents a partial answer for the well-known problem proposed by S.S.
Chern, Problem 2. For further research directions, we address a couple of open
problems; namely Problem 3 and Problem 4.
