How does metric give complete information about its space?

[luinthoron]

Hello,

I've been struggling with the so often spoken idea that a metric tensor gives you all necessary information about the geometry of a given space. I accept that from the mathematical point of view as every important calculation (speaking as a physicist with respect to GTR rather than differential geometry itself) comes down to the metric. However, that is not enough for me. I would like to be able to come up with an aswer that doesn't rely on pure mathematics.

I guess my question boils down to this. How can I know from the validity of Pythagoras' theorem on a given 2D plane that the said plane is flat?

To me, metric tensor is essentially local measuring. The problem is I don't see the connection between this local measurment and saying that, for example, a surface is either flat or curved.

I would appreciate your advice in this. Thank you.

 

[jedishrfu]

The best example is perhaps drawing right triangles on a sphere with say one point at the north pole, one leg along a longitude line and one leg along a latitude line. The triangle defined in this instance is a right triangle drawn on a sphere.

You can see that the pythagorean theorem fails unless the triangle is very small relative to the radius of the sphere then it is approximately true. Historically Gauss ran into this problem while trying to survey a large tract of hilly land and the surveyed pieces didn't fit together precisely.

http://www.surveyhistory.org/carl_friedric.htm

The proper metric for the spherical surface could be used to compute the length of the third leg and so it defines the curvature of the surface.

 

[WWGD]

I'm not sure I understand your question; the/a definition of flatness is that the Pythagorean theorem
is satisfied; this is often how a flat metric is defined. What other definition do you have in mind?

 

[Matterwave]

From another perspective. The metric allows you to find distances. Distances allow you to find geodesics (curves of extremal distance). With your geodesics, you can come up with an equation of geodesic deviation (two initially parallel geodesics either remain parallel, converge, or diverge) and that geodesic deviation is a very good measure of the curvature of the manifold. Is that satisfactory?

 

[George Jones]

The metric (tensor) does not determine the global geometry of a space, but it does, if a metric compatible connection is used, determine the local curvature of a space.

To me, metric tensor is essentially local measuring. The problem is I don't see the connection between this local measurment and saying that, for example, a surface is either flat or curved.

I think that you are confusing intrinsic and extrinsic curvature.

First, take $\mathbb{R}^3$ with the standard metric $ds^2 = dx^2 + dy^2 + dz^2$, and consider $\mathbb{R}^2$ to be a 2-dimensional subspace of constant $z$, and with metric $ds^2 = dx^2 + dy^2$. This metric has zero curvature, and thus this two dimensional subspace is intrinsically (locally) flat. It also has zero extrinsic curvature as a surface in $\mathbb{R}^3$.

Next, take $\mathbb{R}^3$ with the standard metric, and consider a 2-dimensional surface that is a cyllinder with the x-axis as its axis of symmetry. This 2-dimensional surface also has metric $ds^2 = dx^2 + dy^2$, which has zero curvature, and thus this two dimensional subspace is intrinsically (locally) flat. It does, however, have non-zero extrinsic curvature as a surface in $\mathbb{R}^3$.

One way that helps to visualize the flatness of the surface of a cylinder is to imagine a rolled up sheet of graph paper. Even when rolled up, the lines on the graph paper don't "bend" either towards or away from each other.

 

[luinthoron]

Thank you for your posts. The thing is, I am trying to introduce the idea of curvature to high school students and I don't want to drown them in mathematics. The remark about Gauss and the actual mesuaring from jedishrfu seems to be in that direction. I'll dig more into it.

 

[Matterwave]

Introducing this to high school students...I'm sort of doubtful that they will even quite understand what a metric is in that case (I certainly didn't when I was in high school). Perhaps a more pictorial presentation would be more useful? I think in that case, geodesic deviation is also quite useful. You can show how on a sphere the great circles starting from the equator going north (parallel) intersect at the north pole, unlike on a plane where parallel lines never intersect.

 

[atyy]

Along the lines of jedishrfu's #2, you could try looking at Weinberg's "Gravitation and Cosmology". Figure 1.1: Is Middle Earth flat?

Alternatively, he suggests looking at the distance tables given by airlines.

The answer can be obtained with Weinberg's Eq 1.1.4.

 

[WWGD]

Hello,

I've been struggling with the so often spoken idea that a metric tensor gives you all necessary information about the geometry of a given space. I accept that from the mathematical point of view as every important calculation (speaking as a physicist with respect to GTR rather than differential geometry itself) comes down to the metric. However, that is not enough for me. I would like to be able to come up with an aswer that doesn't rely on pure mathematics.

I guess my question boils down to this. How can I know from the validity of Pythagoras' theorem on a given 2D plane that the said plane is flat?

To me, metric tensor is essentially local measuring. The problem is I don't see the connection between this local measurment and saying that, for example, a surface is either flat or curved.

I would appreciate your advice in this. Thank you.

I don't know if this is the type of answer you are looking form, but two isometric manifolds; those in which, in a certain sense, the metric tensor is preserved, have similar notions of angle, distance, etc. The metric tensor, as an inner-product gives rise to the notions of angle and length, and the length itself can recover the original topology of the manifold.

 

[Incnis Mrsi]

The question is not mathematics, but a piece of poorly considered philosophy. There are many different geometries: Euclidean, affine, projective, pseudo-Euclidean, symplectic, metric, differential, Riemannian, pseudo-Riemannian, some others exist also (including such peculiar one as non-commutative), and general topology also can be considered a special flavour of geometry.

Riemannian geometry is useful for the Earth’s surface and, possibly, for stationary solutions in general relativity. But Riemannian geometry doesn’t work in the spacetime, where pseudo-Riemannian geometry rules. Pseudo-Euclidean geometry is good to describe an interstellar travel or the motion of an electron in a cyclotron, but is of no value for physical cosmology. Also there isn’t necessarily a consensus where geometry ends and “fields in the space” start. For instance, Maxwell’s field admits a geometric description in terms of the connection on a linear bundle.

 

[mathwonk]

millman and parker's book, "geometry a metric approach", gives on page 237 a proof that the euclidean parallel postulate is equivalent to the pythagorean theorem in a neutral geometry.

http://books.google.com/books?id=Kp...ce=gbs_ge_summary_r&cad=0#v=onepage&q&f=false


curvature is a local property, so to know you are in the euclidean plane you need also some global hypothesis. If you restrict to surfaces which have constant curvature, there are three types, zero, positive and negative curvature. In each case, normalizing curvature to be say 1, -1, or 0, there are three universal such surfaces which cover all others. E.g. the cylinder has curvature zero and is covered by the euclidean plane.

this theory of surfaces of constant curvature as quotients of universal ones is explained nicely in stillwell's book: geometry of surfaces.

https://www.amazon.com/dp/0387977430/?tag=pfamazon01-20


If you look at euclid's axioms you will notice there are some that can be interpreted in either a local or global way. e.g. the axiom that every line segment can be arbitrarily extended to a line, does not say whether the resulting line is infinite or finite, i.e. does not say whether the extended line loops back on itself and repeats the same loop over and over, or whether it extends forever without covering the same points again. on the cylinder some of the extended "lines" are finite, whereas on the euclidean plane all are infinite.

 

[lavinia]

I like to explain geometric flatness with paper. First make people see that paper is remarkable because you can not stretch it or shrink it. So if you bend a piece of paper, none of the lengths or angles are changed. So the Pythagoean theorem still holds which means that a bent piece of paper is still flat. In particular a cylinder made from rolling up a piece of paper is flat. So is a Mobius band made from a strip of paper. Next have your students try to imagine rolling a piece of paper into a sphere.They will intuitively feel the paper creasing and crinkling and see from this that one would have to warp the paper to cover the sphere. But warping requires stretching and paper does not stretch. Thus this warping means that the sphere is not flat. This pedagogy can be extended a lot further without any formal differential geometry.