Explain it geometrically and tell me how to define these terms mathematically (I am looking for expression).
*... tell me how to define these terms mathematically...*
Lu, Zhiqin. "Normal scalar curvature conjecture and its applications." Journal of Functional Analysis 261.5 (2011): 1284-1308.
From the paper The famous normal sca;lar curvature: "Conjecture. Let f : Mn → Mn+m(c) be an isometric immersion, where Mn+m(c) is a real space form of constant sectional curvature c. Then ρ ≤ ∥H∥ 2 − ρ ⊥ + c, where ρ is the normalized scalar curvature (intrinsic invariant) and ρ ⊥ is the normalized normal scalar curvature (extrinsic invariant)." So
normalized scalar curvature \rho is intrinsic invariant. normalized normal scalar curvature ρ^⊥ is complement of \rho and it is extrinsic invariant.
they are not same They have relation with the following:
$$ {\rho}_N = \frac{2}{n(n-1)} \sqrt{K_N} $$
where $\rho_N$ is the normalized scalar normal curvature and $K_N$ is the scalar normal curvature of $M^n $.
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