Manifold and Metric: Answers to Your Questions

Such a manifold is often called a topological manifold.If it exists, what is its name?It is commonly known as a topological manifold. It is a type of manifold that does not have a metric structure. In summary, a manifold does not necessarily have a metric and if it does not, it is called a topological manifold.
  • #1
princeton118
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Does a manifold necessarily have a metric?
Does a manifold without metric exist? If it exists, what is its name?
 
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  • #2
princeton118 said:
Does a manifold necessarily have a metric?
Does a manifold without metric exist? If it exists, what is its name?

Bob. :)
 
  • #3
princeton118 said:
Does a manifold necessarily have a metric?

Well, you posted this in the General Astronomy forum but I will interpret your question to concern the theory of manifolds in mathematics. With that assumption, no, in general, smooth manifolds need not be provided with any Riemannian (or Lorentzian) metric. There are in fact several intermediate levels of structure between the basic notion of a smooth manifold and the notion of a Riemannian (or Lorentzian) manifold.

princeton118 said:
Does a manifold without metric exist?

Any smooth manifold which has not been provided with a metric tensor in the sense of Riemannian (or Lorentzian) geometry.
 

1. What is a manifold?

A manifold is a mathematical space that locally resembles Euclidean space. In other words, it is a topological space that is smooth and can be described using coordinates.

2. What is the dimensionality of a manifold?

The dimensionality of a manifold is the number of coordinates needed to describe a point on the manifold. For example, a 2-dimensional manifold would require two coordinates, such as latitude and longitude on the surface of a sphere.

3. What is a metric on a manifold?

A metric on a manifold is a way of measuring distances between points on the manifold. It defines the notion of distance and angle on the manifold, similar to how the Pythagorean theorem defines distances in Euclidean space.

4. What is the relationship between a manifold and a metric?

A metric is an essential component of a manifold. It allows us to define distances and angles on the manifold, which in turn allows us to study the properties and geometry of the manifold.

5. How are manifolds and metrics used in physics?

Manifolds and metrics are used extensively in physics, especially in the fields of relativity and quantum mechanics. They are used to describe the geometry of space and time, and to study the behavior of particles and forces within this space. They are also used in various other areas of physics, such as fluid dynamics and thermodynamics.

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