Good treatments on the differential geometry on surfaces.

Thread starter center o bass
Start date Sep 11, 2013
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Differential Differential geometry Geometry Surfaces

In summary, differential geometry on surfaces is a branch of mathematics that studies the properties of curves and surfaces in three-dimensional space. It has many practical applications in fields such as physics, engineering, and computer graphics. Some common techniques used in this field include coordinate systems, vector calculus, and differential forms. It is also related to other areas of mathematics such as topology and algebraic geometry. Current research topics in differential geometry on surfaces include the study of minimal surfaces and the development of new techniques for solving geometric problems on curved surfaces, as well as the use of computer algorithms in discrete and computational geometry.

Sep 11, 2013

center o bass

Hi! I'm trying to read up on the subject of hypersurfaces related to GR; First and second fundamental form, Theorema Egregium etc.. Does anyone know any good treatments? (Books or notes)

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Sep 13, 2013

mathwonk

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heres a nice free set of notes:

http://www.math.uga.edu/%7Eshifrin/ShifrinDiffGeo.pdf

Last edited by a moderator: May 6, 2017

1. What is differential geometry on surfaces?

Differential geometry on surfaces is a branch of mathematics that studies the properties of curves and surfaces in three-dimensional space. It uses concepts from calculus, algebra, and geometry to understand the geometric properties of surfaces and their relationship to curvature, distance, and other geometric quantities.

2. How is differential geometry on surfaces used in real-world applications?

Differential geometry on surfaces has many practical applications, including in the fields of physics, engineering, and computer graphics. For example, it is used to study the shape of three-dimensional objects, to understand the behavior of light and sound waves, and to design efficient and accurate computer algorithms for rendering 3D images.

3. What are some common techniques used in differential geometry on surfaces?

Some common techniques used in differential geometry on surfaces include coordinate systems, vector calculus, and differential forms. These tools are used to analyze the geometric properties of surfaces, such as curvature, length, and area, and to solve differential equations that describe the behavior of curves and surfaces.

4. How does differential geometry on surfaces relate to other branches of mathematics?

Differential geometry on surfaces has connections to many other areas of mathematics, including topology, algebraic geometry, and differential equations. It also has applications in physics and engineering, where it is used to study the shape and behavior of physical objects and systems.

5. What are some current research topics in differential geometry on surfaces?

Some current research topics in differential geometry on surfaces include the study of minimal surfaces, which are surfaces with the smallest possible surface area, and the development of new techniques for solving geometric problems on curved surfaces. Other areas of active research include the study of discrete and computational geometry, which uses computer algorithms to analyze and manipulate geometric objects.