Difference between "intrinsic"and "parametric" curvature

In summary, if all the coefficients in the first fundamental form of a surface are constants, the surface is not curved. However, if the coefficients are not all constants, it depends on the intrinsic curvature of the surface. To distinguish between parametric and intrinsic curvature, you can calculate the Riemann curvature tensor. If it is 0, then the surface is flat. In cases where reparameterization can eliminate the curvature, there are methods such as normal coordinates and isothermal coordinates that can help find Cartesian coordinates. In higher-dimensional manifolds, the Riemann curvature tensor is used to measure intrinsic curvature.
  • #1
hideelo
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I am studying differential geometry of surfaces. I am trying to understand some features of the first fundamental form. The first fundamental form is given by

ds2 = αijdxidxj

Now if the αijs are all constants (not functions of your variables) then I think (correct me if I'm wrong) that the surface is not curved. However if they are not then it depends. I am making the following distinction (which might not be legitimate) between what I am calling parametric curvature and intrinsic curvature. Let me give examples of the two.

The plane ℝ2 can be parametrized with polar coordinates in which case ds2 = dr2 + r22. However although the coefficients are not all constants, it is only the coordinates which are curved and not the surface. Most importantly, with the right reparametrization the curvature will go away.

On the other hand the sphere S2 with spherical coordinates has as its first fundamental form ds2 = dΦ2 + cos2θ dθ2. In this case the curvature is clearly a property of the surface and is intrinsic and (I'm guessing) will not disappear with simply reparameterizing.

My questions are:

1 How can I distinguish between these?
2 In such cases where reparameterization will help eliminate the curvature, is there a simple way to generate this uncurved parameterization?

TIA
 
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  • #2
I would suggest you do not use the term "parametric curvature". This is not a standard terminology, and it suggests the presence of curvature when there is none. A flat manifold is flat.

1) You can calculate the Riemann curvature tensor. If it is 0, then the manifold is actually flat.

2) There's no "general rule" you can find for transforming an arbitrarily complex curvilinear coordinate system back into Cartesian coordinates that I am aware of.
 
  • #3
Since you're specifically studying the differential geometry of surfaces (I assume 2-dimensional surfaces in ##\mathbb{R}^3##?), then you will soon learn about the second fundamental form, which measures the extrinsic curvature of the embedding. And I assume shortly after that you will learn about the Gauss curvature and the Theorema Egregium.

The Gauss curvature measures the intrinsic curvature of a 2-dimensional surface. Therefore, you can find "Cartesian" coordinates if and only if the Gauss curvature vanishes.

Actually finding such coordinates typically requires a bit of guessing, although in 2 dimensions there are some methods, such as normal coordinates and isothermal coordinates, that should give you a Cartesian system if your surface is intrinsically flat.

The Riemann curvature tensor comes into play in higher-dimensional manifolds. In 2 dimensions, you require only one quantity (the Gauss curvature) to measure the intrinsic curvature; but in higher dimensions, you require ##d^2 (d^2-1)/12## independent quantities, which are organized into the Riemann curvature tensor. The Riemann curvature tensor vanishes if and only if a manifold is intrinsically flat (in which case, it will be possible to find local Cartesian coordinates).
 
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What is the difference between intrinsic and parametric curvature?

Intrinsic curvature refers to the curvature of a surface or space itself, independent of any external reference frame. It is a measure of how a surface curves or bends in its own space. Parametric curvature, on the other hand, is defined in relation to an external reference frame and is dependent on the parameters used to describe the surface.

How are intrinsic and parametric curvature calculated?

Intrinsic curvature is calculated using mathematical concepts such as Gaussian curvature and mean curvature, which are based on the behavior of curves and surfaces within the space itself. Parametric curvature is calculated using the parameters that describe the surface, such as curvature along a specific direction or at a specific point.

What are some examples of intrinsic and parametric curvature in real-world applications?

Intrinsic curvature can be seen in the shape of a sphere, where every point on the surface has the same intrinsic curvature. Parametric curvature can be seen in the curvature of a road, where the curvature changes along the length of the road and is dependent on the parameters used to describe it, such as the angle of the curve or its radius.

Why is understanding intrinsic and parametric curvature important in science?

Intrinsic and parametric curvature play a crucial role in fields such as mathematics, physics, and engineering. They help us understand the behavior of surfaces and spaces and how they can be described and manipulated. This understanding is essential in fields such as geometry, cosmology, and computer graphics.

Can intrinsic and parametric curvature be measured or observed?

Intrinsic curvature can be measured or observed through various mathematical techniques, such as calculating the curvature of a surface using differential geometry. Parametric curvature can also be measured or observed by analyzing the parameters used to describe a surface and how they change. Both types of curvature can be visually observed in real-world objects and surfaces.

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