Differential Geometry general question

In summary, if two surfaces are parameterized by X and Y and their coefficients of the First Fundamental Form are the same, then the map X(Y^-1) is an isometry. However, if two parameterizations have different coefficients, this does not necessarily mean that the surfaces are not isometric. In the case where two parameterizations have different coefficients but the same Gaussian curvature, it is necessary to prove that the surfaces are not isometric in order to disprove the converse of Gauss's Great Theorem.
  • #1
InbredDummy
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Ok, in general, I know that if the coefficients of the First Fundamental Form agree for two surfaces parameterized by X and Y, the the map X(Y^-1) is an isometry, or the two surfaces are isometric.

I also know that if two parameterizations don't have the same coefficients, this does not imply that the two surfaces are not isometric.

So i have two parameterizationsthat have different coefficients of the FFF (first fundamental form) but have the same Gaussian curvature. I need to prove that the two surfaces are not isometric. (ie i need to prove that the converse of Gauss's Great Theorem is false).
 
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bump bump any help?
 

FAQ: Differential Geometry general question

What is differential geometry?

Differential geometry is a branch of mathematics that deals with the study of curves, surfaces, and other geometric objects using techniques from calculus and linear algebra. It is used to study the properties of smooth and curved spaces, such as in relativity and physics.

What are some applications of differential geometry?

Differential geometry has many applications in various fields, including physics, engineering, computer graphics, and robotics. It is used to study the shape and curvature of surfaces, to analyze the behavior of particles and waves in curved spaces, and to develop algorithms for 3D modeling and motion planning.

How is differential geometry related to other branches of mathematics?

Differential geometry is closely related to other branches of mathematics, such as differential equations, topology, and algebraic geometry. It also has connections to physics, as it is used to describe the geometry of space and time in general relativity.

What are some important concepts in differential geometry?

Some important concepts in differential geometry include manifolds, which are generalizations of curves and surfaces to higher dimensions, and tensors, which are mathematical objects that describe how quantities change in different directions. Other important concepts include curvature, geodesics, and differential forms.

How is differential geometry used in machine learning?

Differential geometry plays a crucial role in machine learning, particularly in the field of deep learning. It is used to develop geometric models for high-dimensional data, to analyze the behavior of neural networks, and to study the properties of optimization algorithms. Differential geometry also has applications in computer vision and natural language processing.

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