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

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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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