What is the regularity condition in the definition of a regular surface?

[drnickriviera]

Hello everyone,

I'm just getting into differential geometry at the moment and I am confused about one of the conditions in the definition of a regular surface. It is the regularity condition. I'll include the whole definition here for the sake of completeness.

A subset $S\subset \mathbb{R}^3$ is a regular surface if, for each $p\in S$, there exists a neighborhood $V$ in $\mathbb{R}^3$ and a map $\mathbf{x}:U\rightarrow V\cap S$ of an open set $U\subset\mathbb{R}^2$ onto $V\cap S\subset\mathbb{R}^3$ such that

1. $\mathbf{x}$ is infinitely differentiable.

2. $\mathbf{x}$ is a homeomorphism.

3. (The regularity condition.) For each $q\in U$, the differential $d\mathbf{x}_q:\mathbb{R}^2\rightarrow\mathbb{R}^3$ is one-to-one.

(From Differential Geometry of Curves and Surfaces, Do Carmo, 1976)

Now, I feel like I must have missed something at some point prior to studying this, because I really am not sure what the differential $d\mathbf{x}_q$ represents or how to calculate it. The instructor of the class gave an equivalent condition that makes more sense, but for homework it is necessary to prove that condition and so I have to understand what the third condition means. Can anyone help me here?

 

[lavinia]

Hello everyone,

I'm just getting into differential geometry at the moment and I am confused about one of the conditions in the definition of a regular surface. It is the regularity condition. I'll include the whole definition here for the sake of completeness.

A subset $S\subset \mathbb{R}^3$ is a regular surface if, for each $p\in S$, there exists a neighborhood $V$ in $\mathbb{R}^3$ and a map $\mathbf{x}:U\rightarrow V\cap S$ of an open set $U\subset\mathbb{R}^2$ onto $V\cap S\subset\mathbb{R}^3$ such that

1. $\mathbf{x}$ is infinitely differentiable.

2. $\mathbf{x}$ is a homeomorphism.

3. (The regularity condition.) For each $q\in U$, the differential $d\mathbf{x}_q:\mathbb{R}^2\rightarrow\mathbb{R}^3$ is one-to-one.

(From Differential Geometry of Curves and Surfaces, Do Carmo, 1976)

Now, I feel like I must have missed something at some point prior to studying this, because I really am not sure what the differential $d\mathbf{x}_q$ represents or how to calculate it. The instructor of the class gave an equivalent condition that makes more sense, but for homework it is necessary to prove that condition and so I have to understand what the third condition means. Can anyone help me here?

X is a vector of three functions. At each point in U, each of these three functions has a differential. dXq is the vector of these differentials at the point,q.

 

[drnickriviera]

All right, and thanks for your response, but what is the differential of each function? Is it equivalent to the total derivative? Also, what does one-to-oneness imply for $d\mathbf{x}_q$? Is it the regular function definition for one-to-one?

 

[lavinia]

All right, and thanks for your response, but what is the differential of each function? Is it equivalent to the total derivative? Also, what does one-to-oneness imply for $d\mathbf{x}_q$? Is it the regular function definition for one-to-one?

the differential of a differentiable function maps tangent vector to tangent vectors. Thinking of derivatives as maps on tangent spaces is essential for understanding multivariate calculus. In standard Cartesian coordinates the differential is just the Jacobian matrix viewed as a linear map.

 

[blue_raver22]

Dear student,

The regularity condition in the definition of a regular surface ensures that the surface is smooth and does not have any sharp edges or corners. It means that for every point on the surface, there exists a tangent plane that is unique and well-defined. This is important because it allows us to define the concept of curvature and study the behavior of the surface in a smooth and predictable manner.

The differential d\mathbf{x}_q represents the linear transformation between the tangent plane at the point q on the surface and the tangent plane at the corresponding point on the parameter space (in this case, the open set U\subset\mathbb{R}^2). This is why the regularity condition states that this differential must be one-to-one, meaning that it maps distinct points on the parameter space to distinct points on the surface.

To calculate this differential, you can use the chain rule and the fact that \mathbf{x} is an infinitely differentiable map. This will allow you to express the differential as a Jacobian matrix, which represents the linear transformation between the two tangent planes.

I hope this helps clarify the regularity condition for you. Keep up the good work in your studies of differential geometry!