no differenceeee

*... 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 → Mn+m(c) be an isometric immersion, where Mn+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 $.

Actually $K_N$ is written using shaper operator $A$.

Thank you Professor Cenap.