  The notion of different kind of algebraic Casorati curvatures are introduced.
Some results expressing basic Casorati inequalities for algebraic Casorati
curvatures are presented. Equality cases are also discussed. As a simple
application, basic Casorati inequalities for different $\delta $-Casorati
curvatures for Riemannian submanifolds are presented. Further applying these
results, Casorati inequalities for Riemannian submanifolds of real space forms
are obtained. Finally, some problems are presented for further studies.
