When is a surface a manifold

A manifold is a topological space that is locally Euclidean i. To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat. One of the goals of topology is to find ways of distinguishing manifolds.

For instance, a circle is topologically the same as any closed loop, no matter how different these two manifolds may appear. Similarly, the surface of a coffee mug with a handle is topologically the same as the surface of the donut, and this type of surface is called a one-handled torus. As a topological space , a manifold can be compact or noncompact, and connected or disconnected. Commonly, the unqualified term "manifold"is used to mean " manifold with boundary. However, an author will sometimes be more precise and use the term open manifold for a noncompact manifold without boundary or closed manifold for a compact manifold with boundary.

If a manifold contains its own boundary, it is called, not surprisingly, a " manifold with boundary. The concept can be generalized to manifolds with corners. By definition, every point on a manifold has a neighborhood together with a homeomorphism of that neighborhood with an open ball in. In addition, a manifold must have a second countable topology. Unless otherwise indicated, a manifold is assumed to have finite dimension , for a positive integer.

Smooth manifolds also called differentiable manifolds are manifolds for which overlapping charts "relate smoothly" to each other, meaning that the inverse of one followed by the other is an infinitely differentiable map from Euclidean space to itself. Manifolds arise naturally in a variety of mathematical and physical applications as "global objects. From this, we can derive similar formulas using the same similar triangle argument as the previous example.

The symmetric equations can also be found for the "south" pole. To actually calculate things like distance on a manifold, we have to introduce a few concepts. For a 2D manifold embedded in 3D , this would be a plane.

Figure 7 shows a visualization of this on a manifold. It can however look like this when it is embedded in a higher dimension space like it is here for visualization purposes e. Manifolds don't need to even be embedded in a higher dimensional space recall that they are defined just as special sets with a mapping to Euclidean space so we should be careful with some of these visualizations.

However, it's always good to have an intuition. Let's try to formalize this idea in two steps: the first step is a bit more intuitive, the second step is a deeper look to allow us to perform more operations. So far so good, this is just repeating what we had in Figure 4.

Basically, this just defines a curve that runs along our manifold. Now we want to imagine we're walking along this curve in the local coordinates i. Thus, the velocity is just the instantaneous rate of change of our position vector with respect to time. Tangent vectors as velocities only tell half the story though because we have a tangent vector specified in a local coordinate system but what is its basis? Recall a vector has its coordinates an ordered list of scalars that correspond to particular basis vectors.

So understanding how the tangent spaces relate between different points and potentially charts on a manifold is important. Note the introduction of partial derivatives and summations in the third line, which is just an application of the multi-variable calculus chain rule. We can see that by introducing this test function and doing our little trick we get the same velocity as Equation 7 but with its corresponding basis vectors. Okay the next part is going to be a bit strange but bear with me.

We're going to take the basis and re-write it like so:. But it's important to remember that when we're using this notation, implicitly there is a chart behind it.

We know it's some test function that we used, but it was arbitrary. And in fact, it's so arbitrary we're going to get rid of it! Remember, a vector space doesn't need to be our usual Euclidean vectors, they can be anything that satisfy the vector space properties, including differential operators! A bit mind bending if you're not used to these abstract definitions. Now that we have a basis for our tangent vectors, we want to understand how to change basis between them.

Now we want to look at how we can convert from a tangent space in one chart to another. After some wrangling with the notation, we can see the change of basis is basically just an application of the chain rule.

In matrix notation, we would get something like:. For those of you who understand tensors if not read my previous post Tensors, Tensors, Tensors , the tangent vector transforms contravariantly with the change of coordinates charts , that is, it transforms "against" the transformation of change of coordinates.

A "with" change of coordinates transformation would be multiplying by the inverse of the Jacobian, which we'll see below with the metric tensor. So after all that manipulation, let's take a look at an example on our sphere to make things a bit more concrete. Even though we now know how to find tangent vectors at each point on a smooth manifold, we still can't do anything interesting yet! To do that we'll have to introduce another special tensor called -- you guess it -- the metric tensor!

In particular, the Riemannian metric tensor 1 is a family of inner products:. Note that this is a family of metric tensors, that is, we have a different tensor for every point on the manifold. The implications of this is that even though each adjacent tangent space can be different the manifold curves therefore the tangent space changes , the inner product varies smoothly between adjacent points.

A real, smooth manifold with a Riemannian metric tensor is called a Riemannian manifold. Intuitively, Riemannian manifolds have all the nice "smoothness" properties we would want and makes our lives a lot easier. That is, we're going to define our family of Riemannian metric tensors using the metric tensor from the embedded Euclidean space. This guarantees that we'll have this nice smoothness property because we're inducing it from the standard Euclidean metric in the embedded space.

Starting from Equation So here, we're simply playing around with the chain rule to get the final result. The basis is in our embedded space using a similar notation to our local coordinate tangent basis.

In fact, we could derive the Equation 23 in the same way as Equation 8 using the "velocity" idea. Whether in our local tangent space or in the embedded space, they're the same vector i. Now that we know how to convert between the tangent spaces, we can calculate what our Euclidean metric tensor i. So we can see that the induced inner product is nothing more than the matrix product of the Jacobian with itself of the mapping from the local coordinate system to the embedded space.

Notice that this multiplication by this Jacobian is actually a "with" basis transformation, thus matching the fact that the metric tensor is a 0, 2 covariant tensor. Now with the metric tensor, you can compute all kinds of good stuff like the length or angle or area.

I won't go into all the details of how that works because this post is getting super long. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search.

A surface is simply a two-dimensional manifold. However, some subtlety arises if you distinguish "topological manifolds" Hausdorff, second-countable, locally Euclidean topological spaces , "differentiable manifolds" with additional global differential structure , and "smooth manifolds" where the structure is smooth, i. See, for example, this. Usually a surface is defined as a two-dimensional topological manifold; the answer might depend on context, in particular your definitions of "surface" and "smooth manifold".

Any manifold of higher or lower dimension would not be a surface. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.