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dg.differential geometry – Calculation of the denote curvature below a regular perturbation

Let $X: M^n to N^{n+1}$ breathe a Riemannian immersion. Write $g, A, nu, H$ for the primary elementary figure, second elementary figure, Gauss map and denote curvature of $X$ respectively. Consider the regular pertubation $X_u: M to N$ outline by $X_u := X + unu$, the place $u$ is flush. For diminutive sufficient $u$, we’ve that $X_u$ is an immersion.

In the particular illustration the place the dimension of the hypersurface is $n=2$ and the ambient manifold is Euclidean $N = mathbb{R}^{n+1}$, Nicolaos Kapouleas provides a calculation of the quadratic and better organize phrases denote curvature of $X_u$ in ‘Complete Constant Mean Curvature Surfaces in Euclidean Three-Space’, Appendix C.

Is there a reference or calculation of the denote curvature of $X_u$ (or its linear/quadratic phrases) within the extra common circumstances of when the dimension $n$ isn’t equal to $2$ and/or when the ambient manifold $N$ isn’t Euclidean?

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