Regular surface in Rn and Inverse image of a regular value

Regular surface in Rⁿ and Inverse image of a regular value

Abstract

For the surface, first we know that the surface is a two-dimensional to three-dimensional mapping, which can be described in mathematical language as follows:
r：U → V，U⊂R²，V⊂R³
There are countless curves passing through a point P on the surface. These curves are regular curves in the neighborhood of P, and these curves do not self intersect in the neighborhood of P, so this P point is a regular point on the surface. If all points on the surface are regular points, this surface is a regular surface.Next, I will extend this definition to R space, and discuss the inverse image of a regular value.

Prepare

Inverse Function Theorems on Manifolds

Let X and Y are smooth mianfolds, a smooth mapping f: X→Y.
If the tangent mapping dfx₀ of f at x₀ is invertible linear mapping,
Then ∃ V ⊂ X, X₀ ∈ V and W ⊂ Y, f(X₀) ∈ W, such that f|_V= V ≃ W is Differential homeomorphism.
And, for any x ∈ V, y = f(x), then d(f⁻¹)y = (dfx)⁻¹.

critical value and regular value

Let F: Rⁿ → R^m be a differentiable mapping of an open set U of Rⁿ.

1. A point p ∈ U is defined to be a critical point of F if the differential dF_p: Rⁿ