The discussion revolves around the construction of a manifold locally using a metric, specifically focusing on the use of inner products of basis vectors to achieve this. Participants explore the implications of different types of inner products and the relationship between the manifold and its embedding in higher-dimensional spaces.
Participants express differing views on the clarity of the original question and the appropriate methods for constructing a metric. There is no consensus on the best approach, and multiple perspectives on the use of inner products and the nature of the manifold are presented.
Participants highlight the importance of understanding the nature of the basis vectors and the context of the manifold, indicating that assumptions about the manifold's properties and the embedding space are critical to the discussion.
Let me shorten it a little further, can I take the inner product of basis vectors and form a suitable metric?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?
Your question is far too vague to be answered.
Inner product is scalarfresh_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.
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?dsaun777 said:Let me shorten it a little further, can I take the inner product of basis vectors and form a suitable metric?
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?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?
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?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.