91Ó°ÊÓ

A regular surface in the context of differential geometry is a smooth and well-behaved surface. Mathematically, it's a subset of \( \mathbb{R}^{3} \) that can be described by a parametric equation \( \mathbf{X}(u_1, u_2) \). This means there is a smooth function \( \mathbf{X}: U \subseteq \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) that is injective, meaning no two values in the domain map to the same point on the surface.
When looking at a map like the one given in the exercise, which scales the coordinates by a positive constant \(c\), we want to ensure that the transformed surface remains regular. Because the map is linear and scaling does not affect the smoothness or injectivity, the new surface formed by applying \(c\) is also regular.

The Gaussian curvature (K) of a surface is a way to measure its intrinsic curvature at a given point. It takes into account how the surface curves in different directions from that point.
It's derived from the first and second fundamental forms of the surface, which capture the surface’s metric and shape, respectively. Notably, Gaussian curvature is an intrinsic property, meaning that it does not change if the surface is bent without stretching. However, when the surface is scaled by a factor of \(c\), the curvature is affected: the Gaussian curvature \(\bar{K}\) of the transformed surface is given by \(\bar{K} = \frac{K}{c^2}\). This means scaling a surface reduces its curvature by the square of the scaling factor.

The mean curvature (H) of a surface represents the average of the curvatures in two orthogonal directions. While Gaussian curvature is a measure of intrinsic curvature, mean curvature gives information about how the surface bends in space.
It's especially important in describing physical phenomena like minimal surfaces (which occur in soap films) or the shape of biological membranes. When the surface is scaled by a factor \(c\), the mean curvature \(\bar{H}\) is transformed by \(\bar{H} = \frac{H}{c}\). Thus, the mean curvature decreases linearly with the scaling factor, which provides critical information when analyzing surfaces under transformations.

A parametric equation provides a way to describe surfaces in \( \mathbb{R}^{3} \) using two parameters, usually denoted as \(u_1 \) and \(u_2 \). These parameters can be thought of as coordinates on a 2D domain that is mapped onto the 3D surface.
For a surface \(S \) defined by \( \mathbf{X}(u_1, u_2)\), the transformation \( F(p) = cp \) results in a new surface \( \bar{S} \) with a parametric equation given by \( \bar{\mathbf{X}}(u_1, u_2) = c \mathbf{X}(u_1, u_2) \). This demonstrates how a similarity transformation impacts the representation of the surface.

The fundamental forms are mathematical objects that capture the metric and curvature properties of a surface.
The first fundamental form involves dot products of tangent vectors and is related to the lengths and angles on the surface. The second fundamental form relates to the surface's bending by examining how the normal vector changes. Together, these forms provide all the necessary information to compute geometric properties like Gaussian and mean curvatures.
When a surface is transformed by scaling, its fundamental forms are adjusted accordingly, ensuring that properties such as Gaussian and mean curvatures can be recalculated for the transformed surface. For example, if the original first and second fundamental forms are denoted as \( E,G,F \) and \( L,M,N \) respectively, the forms for the scaled surface will be affected proportionally by the scaling factor.