| Summary: | In the present paper, we prove that if Laplacian for the warping function of complete warped product submanifold <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi mathvariant="double-struck">M</mi> <mi>m</mi> </msup> <mo>=</mo> <msup> <mi mathvariant="double-struck">B</mi> <mi>p</mi> </msup> <msub> <mo>×</mo> <mi>h</mi> </msub> <msup> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msup> </mrow> </semantics> </math> </inline-formula> in a unit sphere <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">S</mi> <mrow> <mi>m</mi> <mo>+</mo> <mi>k</mi> </mrow> </msup> </semantics> </math> </inline-formula> satisfies some extrinsic inequalities depending on the dimensions of the base <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">B</mi> <mi>p</mi> </msup> </semantics> </math> </inline-formula> and fiber <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msup> </semantics> </math> </inline-formula> such that the base <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">B</mi> <mi>p</mi> </msup> </semantics> </math> </inline-formula> is minimal, then <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">M</mi> <mi>m</mi> </msup> </semantics> </math> </inline-formula> must be diffeomorphic to a unit sphere <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">S</mi> <mi>m</mi> </msup> </semantics> </math> </inline-formula>. Moreover, we give some geometrical classification in terms of Euler–Lagrange equation and Hamiltonian of the warped function. We also discuss some related results.
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