Section 1: Geometry of in

Consider to be a differentiable, time-parameterized curve through . The length of such a curve is defined as

We say is if for all . Now we’ll introduce some definitions to build up to a notion of curvature. Supposing p.a.l., we define and call it the to at time . Intuitively, we define the , denoted to be the magnitude of the time derivative of , Put simply, the curvature tells us how quickly the unit tangent vector of our curve changes! A little more digging will give allow us to draw a clear picture of all of these quantities. Notice that, since , we have This tells us that is necessarily orthogonal to the unit tangent, . We can use this insight to further define the , This gives us a solid basis of geometric descriptors of a given differentiable curve . A depiction of an example , and is shown

Section 2: Geometry of in

Now we can graduate to analyzing the geometry of in . Before we do so, though, we will define the of a map . The differential of at , denoted , is a linear map that takes tangent vectors to curves passing through to the associated tangent vector of the curve induced by , show in Figure . Formally, let be a curve such that and . Then, While this definition may seem confusing, if we take a look at the matrix representation of this linear map, we see that it simply boils down to the jacobian of applied to the vector ,

This becomes clear when we expand the definition of the differential using the chain rule, where by definition and , .

Now we can finally discuss surfaces in , specifically surfaces.

is often called a or a of . This definition establishes the notion that a regular surface can be constructed by stitching together deformations of patches of . Then, by analyzing these deformations we’ll be able to quantify and characterize the geometry of !

Tangent curves and spaces

We’ll now define the notion of a tangent space to a point, a critical geometric object in Differential Geometry. Given a regular surface , we we say is a tangent vector if there exists a curve such that and . A visualization is provided

Figure shows an example of the tangent space to a surface. This vector space actually arises directly from the charts used to construct . This is illustrated in the following lemma:

I’ve omitted the proof, but it arises quite simply from the definitions of the differential and the tangent space. A simple corollary of this result follows too,

Critically, this means we have access to a basis of our tangent space about each arising only from the chart . Now that we’ve established a notion of tangency, we’ll do the same for normalcy.

Normal vector fields and the Gauss map

A vector field on (i.e. a differentiable map ) is called if we have . Furthermore, is called if .

Unitary normal vector fields give rise to the notion of . For example, non-orientable surfaces like the Möbius strip do not admit a global unitary normal vector field. This idea is beatifully illustrated by a painting by M.C. Escher, shown

If it exists, the unitary normal vector field for a surface is called the . Studying the behavior of the Gauss map will lead to a generalization of the notion of curvature for .

The First and Second Fundamental Forms

The first fundamental form (denoted ) and second fundamental form (denoted ) are quadratic forms on the tangent space of surfaces - namely, they take a vector pair to a number in . These two maps describe describe respectively the and geometry of a surface , and will be the subject of our study for the rest of the blog post.

A visualization of the quantities used in the definition can be seen in For me, it is intuitive to think about the matrix representation of . Expanding the definition we see,

where

By looking at the first fundamental form this way, we see that it tells us how much the local chart deforms the patch along the coordinate axes and to get the patch ! We can make this idea concrete by computing path lengths on , where we will see a clear dependence on the first fundamental form.

Let and let , where is a local chart for . Lets compute the length of the path of and : We can apply the chain rule to get in terms of , Plugging back into the path length equation, we get Rewriting as follows, clearly illustrates the fact that determines the distortion of path lengths induced by the chart . Now, we will move on to define and analyze the fundamental form.

Again, we can look at the matrix representation of to get a better idea of what is going on, where is the jacobian of the Gauss map. Some calculation reveals that where,

Again, this sheds some light on what represents: it tells us how the normal vector changes as we move along the surface (specifically with respect to changes in the parameters and in the domain of the local chart ).

The term on the right, , represents the projection of the acceleration of onto the surface normal. This quantity is often referred to as the , as it represents the amount of curvature of that is to the surface at .

:

where the last step follows from the fact that .

From the principal curvatures, the definitions of and curvature arise:

For the rest of the blog we’ll focus on curvature. While mean curvature offers a tool for the study of the extrinsic geometry of a surface, it is out of the scope of this blog post.

To give some grounding for Gaussian curvature: the plane has , the standard 2-sphere has , while the standard pseudosphere has . But a key question arises: It’s not clear how to solve the minimization problem required to compute the principal curvatures. But if we study a little more, we’ll find that it has the ingredients to allow us to compute and directly from a local chart ! Recall

We can directly apply Euler’s Theorem to replace the right hand side with the second fundamental form applied to the initial velocity of , Because this last line only depends on the zero and first-order behavior of , we can replace it with the following, The interpretations of the eigenvalues of a quadratic form as the minimal and maximal values of the quadratic form restricted to the unit sphere tells us that the eigenvalues , of are the principal curvatures:

This gives rise to new equations for the Gaussian and mean curvatures:

Section 3: Gauss’ Theorema Egregium

In this section we’ll walk through Gauss’ , one of the major results of differential geometry. To do so, we first need to establish the definition of and some properties of them.

An example of an isometric surface pair, the plane and the S-surface, is shown in These surfaces are isometric because we can bend, but not stretch, the plane so that it coincides with the S-surface. Such a transformation does not distort the length of paths!

An isometric map between patches of and preserves path lengths. We also have shown that the distortion of such path lengths incurred by the charts and is determined completely by their respective first fundamental forms. From here, it’s not hard to show that since an isometry exists (and path lengths are therefore preserved under this map) the first fundamental forms of the two surfaces must coincide.

Now we are in a position where we can state Gauss’ .

One can show that the Gaussian curvature depends only on the zero and first order behavior of the first fundamental form , and . Thus, can be written as where is some messy function of the first fundamental form and its partial derivatives w.r.t. and . Since can be written as a function of only the first fundamental form, and by the previous theorem isometries preserve the first fundamental form, we can conclude that the Gaussian curvature must be preserved under isometries!

Implications

Gauss’ is a major result of differential geometry, as it sheds light on the distinction between and curvature. Take, for example, an open subset of the plane and roll it into a cylinder. Intuitively (and one can check this by computing ) the lengths of paths are completely unchanged. Since for the plane, Gauss’ tells us that this must mean for the cylinder!

This arises from the fact that Gaussian curvature measures the distortion of a surface from a Euclidean space. Rolling the plane up into a cylinder introduces curvature, but does not stretch or warp the surface in any way. One can see this by playing around with a piece of printer paper. Anything you do to bend or fold the surface that does not introduce any tears or pleats is introducing curvature (and assuming no hard creases are made, such a transformation is an isometry)!

Another huge implication of the is in the field of Cartography. Since for the plane and for the sphere, the surfaces must not be isometric. Therefore, one cannot create a projection of the sphere onto the plane.

Bibliography

Reference