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Casco
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I'm having my first differential geometry course and I can't get the concept.
Casco said:I'm having my first differential geometry course and I can't get the concept.
lavinia said:In Differential Geometry a metric is a length and angle measure on each tangent space that varies smoothly from point to point. Usually this measure is a positive definite inner product but in the theory of relativity it is not positive definite.
Casco said:So, would it be correct to say that a metric gives a way to measure the space-time on a given geometry? Thinking about the special case of General Relativity.
felper said:your course is about manifolds?
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I don't know much about the alternatives, but the books by John M. Lee are so good that I find it very hard to believe that there could be a better option. I'm talking about two books with the titles "Introduction to smooth manifolds" and "Riemannian manifolds: an introduction to curvature". The only problem is that the material is broken up into two books. The latter (which was actually written first) is a pretty short book that contains the material about connections, parallel transport, geodesics and curvature.Casco said:Yes, I think my main problem was the concept, but thanks this I have a better understanding of it. And yes I'm having my first course on Diff. Geometry, the notation is very confusing, so any advice about a good book about Diff. Geo. would be great.
Another question. I have Wald. of General Relativity. Would it be recommendable to read it or should I better let it for other course??
Fredrik said:I don't know much about the alternatives, but the books by John M. Lee are so good that I find it very hard to believe that there could be a better option. I'm talking about two books with the titles "Introduction to smooth manifolds" and "Riemannian manifolds: an introduction to curvature". The only problem is that the material is broken up into two books. The latter (which was actually written first) is a pretty short book that contains the material about connections, parallel transport, geodesics and curvature.
Wald's book is a good GR book that takes the mathematics seriously. It's a good place to read about the most important application of differential geometry. Read it if you're interested in GR. Don't read it if you only want to pass the exam. (Read Lee's books instead).
Singularity said:Hi, ok here is my 2 cents.
Let's just take a normal n-dimensional vector space V to start with. Then a metric, in the differential geometry sense, is a symmetric non-degenerate bilinear form on V. If we define it by g say, then g eats up two vectors and spits out an element of the field over which V is defined. For example g(v,w) = k.
Something for you to think about: What role does the non-degeneracy play? Why do we want it?
Q: What does the smoothness mean here?
Hope this helps!
Casco said:I have heard a little about Lee's books, I will go for them and check them. And the purpose of my Diff. Geometry course is to go later for a GR course. But I think I will let Wald's for later.
So, Do you recommend for a first reading Lee's Introduction to smooth manifold or Riemannian manifolds: an introduction to curvature?
Start with "Introduction to..." and read enough to make sure that you understand manifolds, tangent spaces, and tensor fields. Then you can choose if you want to read more of that book or read the first few chapters of "Riemannian...". (There's nothing in "Introduction to..." about connections, parallel transport, geodesics and curvature. These concepts are essential in GR).Casco said:So, Do you recommend for a first reading Lee's Introduction to smooth manifold or Riemannian manifolds: an introduction to curvature?
A metric, in the scientific sense, is a quantitative measure that is used to assess and compare the characteristics or performance of a system or phenomenon. It is often represented by a mathematical function that maps inputs to outputs.
A measurement is a single value or quantity that is obtained by using a particular metric. In other words, a metric is a tool or method used to obtain measurements, while a measurement is a specific result obtained from using a metric.
An intuitive definition of a metric is a way of quantifying something that is often abstract or difficult to measure. It is a way of assigning meaningful numbers to certain characteristics or behaviors that can help us better understand and compare different systems or phenomena.
A formal definition of a metric is a set of rules or properties that a mathematical function must satisfy in order to be considered a valid metric. These properties include non-negativity, symmetry, and the triangle inequality, among others.
Metrics are commonly used in scientific research to quantify and compare various aspects of a system or phenomenon. They can help researchers make objective and data-driven conclusions, and can also be used to track changes or improvements over time.