# Trying to construct a particular manifold locally using a metric

• I
• dsaun777
In summary: Well forget about the ambient space. What would be the best way of finding a metric if you could only work with the basis vectors of the manifold?This depends on what you want to do. Normally, it is to use a projection onto a coordinate space. See also connections.This depends on what you want to do. Normally, it is to use a projection onto a coordinate space. See also connections.
dsaun777
I am trying to construct a particular manifold locally using a metric, Can I simply take the inner product of my basis vectors to first achieve some metric.

What do you have and where do you want to go to? Where are your basis vectors from? What is the space?

Orodruin
fresh_42 said:
What do you have and where do you want to go to? Where are your basis vectors from? What is the space?

Let me shorten it a little further, can I take the inner product of basis vectors and form a suitable metric?

Yes, if the inner product is one, i.e. positive definite, then it defines angles, lengths and distance. However, manifolds are usually not flat, which is why I wondered what your inner product is.

fresh_42 said:
Yes, if the inner product is one, i.e. positive definite, then it defines angles, lengths and distance. However, manifolds are usually not flat, which is why I wondered what your inner product is.
Inner product is scalar

dsaun777 said:
Let me shorten it a little further, can I take the inner product of basis vectors and form a suitable metric?
Sorry, but it is completely unclear what you mean by this. You have still not answered the question in #2: What do you have and where do you want to go?

Orodruin said:
Sorry, but it is completely unclear what you mean by this. You have still not answered the question in #2: What do you have and where do you want to go?
Assume I am in some surface of a differentiable manifold embedded in a 3 dimensional space. Given that I have some covariant basis vectors along the surface, is it as simple as taking the inner product of basis along the surface to construct a metric? How do you describe the surface in terms of the ambient 3d space using normal surface vectors?

You don't have covariant basis vectors of the surface, only of its tangent space which is a different one at each point. If you want to transport the metric onto the surface, you can either use the outer metric, since it is an embedded surface, or line integrals on the surface. See also connections. The outer metric is more or less worthless, as it does not measure an actual distance, which would be a geodesic.

fresh_42 said:
You don't have covariant basis vectors of the surface, only of its tangent space which is a different one at each point. If you want to transport the metric onto the surface, you can either use the outer metric, since it is an embedded surface, or line integrals on the surface. See also connections. The outer metric is more or less worthless, as it does not measure an actual distance, which would be a geodesic.
Well forget about the ambient space. What would be the best way of finding a metric if you could only work with the basis vectors of the manifold?

## 1. What is a manifold?

A manifold is a mathematical concept that describes a space that locally resembles Euclidean space. It is a topological space that is smooth and can be described by a set of coordinates.

## 2. How is a manifold constructed?

A manifold can be constructed by using a metric, which is a mathematical tool that measures distances and angles in a space. By defining a metric, one can construct a manifold with a specific shape and properties.

## 3. What is the purpose of constructing a manifold locally?

Constructing a manifold locally allows us to understand the geometry and properties of the space in a small region. This can then be extended to the entire space, allowing us to study the manifold as a whole.

## 4. What are some applications of constructing a manifold locally?

Manifolds are used in many fields of science, including physics, engineering, and computer science. Local constructions of manifolds are particularly useful in studying the behavior of systems at a small scale, such as in quantum mechanics or computer simulations.

## 5. Are there different methods for constructing a manifold locally?

Yes, there are various methods for constructing a manifold locally, such as using charts and atlases, defining a Riemannian metric, or using differential equations. The choice of method depends on the specific properties and applications of the manifold being constructed.

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