Explaining the idea of a 2 D surface embedded in 3 D Space.
You will get a (much) better answer than this, but a straightforward reason is that for using extrinsic curvature, we need an extra dimension to put the lower dimensional object "into".
Intrinsic curvature of a surface or manifold can be performed by using math techniques developed by Gauss and Riemann and allows us to do without the extra dimension in curvature calculations.
Obviously everything we see around us is actually in 3D, but we can pretend a sheet of paper is 2D, so looking at a sheet of paper rolled up into a cylinder is looking at a 2D surface in a 3 D space and we can measure directly how much it's curved.
Image Source: Descriptions of Curvature
From looking at these 3 shapes, it's quite obvious that in a 3 D world, we could physically measure all aspects of them, and determine which of them has, in any small region on it, positive curvature, (the sphere), negative curvature, (the "vase" shape) and which has no curvature, (the middle cylinder).
So 2D embedded objects in 3D space are easy to measure.
But we have to describe 4 D objects and events in a 4 D space, so we don't have the "elbow room" of an extra dimension.
So we give up on the visual attempt ( which is why most, if at all depictions of real 4D on TV and books are misleading) and we use expressions such as this:
$${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$
to calculate how much spacetime is curved.
This above is an example using intrinsic curvature, we don't (and actually can't) "lift " ourselves above the space.
We are in 4 D space-time, so to perform calculations in this space, we must use intrinsic curvature math ( differential geometry) as we can't go to a higher dimension.
In slightly more specific terms,you might have read about the metric tensor, which is based around the idea of using basis vector and their components to calculate distances (in 4 dimensions) utilising differential geometry to measure deviations from flat space caused by massive objects.
The other benefit of the intrinsic based metric tensor is that it is applicable in all GR calculations, (invariance).
Let's first clarify the terminology and setting. Let's assume that we have an embedded submanifold $S$ (e.g. a surface) in a Riemannian manifold $M$ (e.g. Euclidean 3-space).
I assume that by intrinsic curvature of $S$, you mean the Gaussian curvature of a surface. For higher dimensional manifolds, this generalizes to the sectional curvature, but this is a little more complicated: it assigns a number to each 2-dimensional subspace of the tangent space, namely the Gaussian curvature of the submanifold (surface) tangent to that plane.
"Extrinsic curvature" could mean several things, but for a hypersurface (e.g. an embedded surface $S$ in Euclidean 3-space), we could summarize by saying that an extrinsic curvature is a quantity defined by the second fundamental form, or equivalently its associated shape operator $B$. If you don't know what these things are, it's ok, you can keep reading. Think of the shape operator as a symmetric matrix depending on $p \in S$. The main "extrinsic curvatures" that are worth considering are:
- The eigenvalues of $B$, called principal curvatures. Note that they are equal to the curvature of curves lying in $S$, seen as curves in $M$.
- The trace of $B$ (maybe divided by the dimension), called the mean curvature, equal to the sum (or average) of the principal curvatures.
For a surface in a 3-dimensional manifold, the Gauss equation says that:
$$ \det B = K_S - K_M$$
where $K_S$ is the Gauss curvature of $S$ and $K_M$ is the sectional curvature in $M$ of the plane tangent to $S$. This equation is probably one of things you're looking for answering your question: it tells you the relation between the second fundamental form (defining the "extrinsic curvatures"), the intrinsic curvature of $S$ and the intrinsic curvature of $M$.
Now let's answer your question more precisely. As you can tell from Gauss equation, if all the extrinsic curvatures are zero, i.e. $B$ vanishes (FYI, in this case one says that $S$ is a totally geodesic submanifold), then the intrinsic curvature of $S$ is equal to the intrinsic curvature of $M$. In particular, if $M$ has zero curvature (i.e. Euclidean 3-space), then a submanifold for which all the extrinsic curvatures are zero also has zero intrinsic curvature.
That being said, if by "the extrinsic curvature" we just signify the mean curvature $\mathrm{tr}(B)$, we are just looking for surfaces with zero mean curvature, these things are called minimal surfaces. Now the question is: are there minimal surfaces that are not totally geodesic (basically, minimal surfaces that are not planes in Euclidean 3-space)? You can guess that the answer is probably "yes, there are plenty", because that's pretty much asking if there are some symmetric matrices $B$ whose trace is zero, but are not the zero matrix. In fact, the "fundamental theorem of surface theory" basically guarantees that there are many examples. You'll see examples by looking for images of "minimal surface" on the web.
I tried to give a complete and detailed answer, I hope it was useful, but if you're looking for the short answer: yes, any minimal surface in Euclidean 3-space that is not a plane has zero mean curvature but nonzero Gaussian curvature. For example, the catenoid:
Best Answer
Oh, that is a really nice question. Thank you for asking :)
Now, let start dissecting your claims and confusions.
First of all, topology is irrelevant here since curvature is a purely local phenomenon. Essentially, curvature is the second derivative of a geometric structure involved. More precisely, curvature comes from a specific expression that exhibits an invariant behavior. This expression, as I said, is built from the second derivatives of some quantity which is thought as a geometric structure on the given space. I deliberately use a very vague language in order to be able to refine my statements for the specific cases that will come shortly.
Secondly, there are two kinds of geometric structures that we can observe on surfaces embedded into $\mathbb{R}^3$ (or, in any other ambient Riemannian manifold: just keep in mind that implicitly we regard $\mathbb{R}^3$ as a Riemannian manifold endowed with the standard Euclidean metric). The first structure is the induced Riemannian metric on the embedded surface (or, more generally, on a submanifold). The second structure is the embedding itself (the way how the surface has been placed into the ambient manifold).
With this said, one can also notice that if we embed our surface (submanifold) into a more general ambient Riemannian manifold, with a Riemannian metric on it, then we actually have the third (or, the zeroth, if you like to count this way) geometric structure involved, the ambient Riemannian metric.
If we now forget about the ambient manifold and the embedding, the induced Riemannian metric will be still there, that is defined as a certain quantity, on the surface (submanifold), but now this surface (submanifold) looks to us as an abstract manifold with some Riemannian metric given on it. We can even move on and embed something else into our new ambient space, which just a minute ago used to be embedded! This Riemannian metric that remains defined on our space is intrinsic in the sense that it does not need to have any external structures anymore.
In this sense, the Riemannian metric on the ambient space is intrinsic to this ambient space, but we refer to this metric as the ambient metric when we look from the perspective of the embedded submanifold. As I said, in the case of $\mathbb{R}^3$, the ambient metric is the standard Euclidean metric.
The embedding itself is also a quantifiable information, which can be differentiated (provided, it is smooth), so that we can obtain expressions of the derivatives of this information. This information is very delicate and hard to reason about, if your view it from the traditional perspective, as a mapping from $\mathbb{R}^m$ into $\mathbb{R}^n$, because it quickly brings us to a coordinate hell, with a lot of technical difficulties blowing up. A smart trick which I learned from Prof. A.Rod Gover, is to use defining functions.
We call a function $\varphi \colon M \to \mathbb{R}$ a (local) defining function for a hypersurface $S$ in an ambient manifold $M$, if (in a neightborhood of some point on $S$) S is the zero locus of function $\varphi$, that is $S = \{ p \in M | \varphi(p) = 0 \}$, and the differential $\mathrm{d}\varphi$ never vanishes at all points on $S$ (within the given neighborhood). This definition can be extended to submanifolds of any co-dimension. Of course, this construction is known for ages.
The traditional way of representing the embedding can be viewed as explicit, and using a defining function is then implicit.
From the first derivative of the defining function we can construct the unit normal along the hypersurface, namely by setting $N := \frac{\mathrm{d}\varphi}{|\mathrm{d}\varphi|}$. It is easy to verify that (up to the sign) the value of $N$ does not depend on the choice of $\varphi$ along $S$. This $N$ is a vector field along $S$, and its ambient covariant derivative ( i.e. with respect to the Levi-Civita connection of the ambient metric) is a tensorial quantity, called the shape tensor $L$ of the embedding: $L := \nabla N$ (I am a little bit sloppy here for the sake of brevity). If you expand the definitions, you will get (a long and ugly) expression in terms of the second derivatives of $\varphi$ and the ambient Riemannian metric.
This shape tensor, up to certain index-juggling identifications, is also known as the second fundamental form. The induced metric on the submanifold in this tradition is referred to as the first fundamental form (of the embedded submanifold).
If you think about the definition of the shape tensor, you will be able to visualize it as a rate of change of the unit normal vector along the surface. This is exactly how you perceive the shape of the embedded surface!
The quantities that depend on the way the surface (submanifold) is embedded are referred to as extrinsic.
In this sense, the shape tensor is the extrinsic curvature of the embedding.
The curvatures of the Riemannian metric, on the other hand, are intrinsic with respect to the manifolds that they are defined on, but we adhere to the convention to refer to the metric induced on the submanifold as the intrinsic metric, and the metric on the ambient manifold as the ambient metric, but this is just a terminological convention.
As you can "feel" the shape by the unit normal's wobbling, the same way one can "feel" the intrinsic curvature from how the geodesics behave within the surface (manifold), regardless of the way the surface (manifold) is embedded, provided the embedding was isometric (that is the intrinsic metric was preserved, in other word the embedding induces the original metric).
To conclude, I will add some comments on your statements.
A 2-cylinder is not necessarily flat, it depends on the embedding.
The curvature can be detected from inside, because, for instance, we can detect the intrinsic metric. Alternatively, we can observe the diverging geodesics.
When you measure angles and distances, you essentially study the intrinsic metric. This can be done on a cylinder too.
When you see a spherical shape, you see the extrinsic curvature. When you study lengths and angles within the surface, you detect the intrinsic curvature.
I hope that my remarks can help to clarify this terminology a little bit.