# Riemannian Geometry We can start by thinking about straight lines. We can think about them several different ways: 1. Shortest distance between two points * In flat euclidean space this just a straight line * In spherical geometry the shortest distance line would be part of a great circle 2. Keep walking in the same direction * This requires that we have a way of "carrying along" our vectors. This is referred to as **parallel transport**. Parallel transport is a way of defined straight lines and characterizing the deviation from curvature. Reimann invented the [Metric](Metric.md) - we can think of this as the "infinitesimal length". It generalizes pythagorus theorem to infinitesimals where we have: $c^2 = \alpha a^2 + \beta b^2 + \gamma ab$ Where $\alpha, \beta, \gamma$ tell us how to convert coordinates ($a$ and $b$) to physical lengths ($c$). Now $\alpha, \beta, \gamma$ can change from place to place in the [Manifold](Manifold.md) (where the manifold is just the space of points, where the space if curved - and of course the curvature could be 0). Now the metric is a [Field](Field.md) on the manifold. It is a kind of field that exists at every point and at every point it answers the question "if I have a little straight line segment how long is it?". We can then add them up to get an entire length. ### Parallel Transport This is simply: we have some vectors, keep them parallel to what they were as you carry them around. Let's spoil why this is so important: parallel transport provides a way to determine if the space we are in is curved. Parallel transport has the nice property that if two vectors start perpendicular (orthogonal) they will remain perpendicular (orthogonal) as you move them. What we may see is that even if we parallel transport a single starting vector along two paths, we may end up with two different final orthogonal vectors (below - we start at $P$, we parallel transport it "straight" to $N$ via the green path, then parallel transport it to $Q$ and then $N$ via the red path). We see that the final vectors at $N$ at not the same, even though we just parallel transported in both cases! This is because *the space is curved*. On a flat plane this would not have happened - we would have ended up with the same vector.  > This is yet another way in which curvature shows up in geometry. It shows up because the angles inside a triangle are different, it shows up because the radius of a circle to it's circumference is different, and now it also shows up in the way that vectors change when you parallel transport them. If you are in a flat space a vector doesn't change at all when parallel transported. In a curved space it changes a lot. This is a reflection of the fact that when you are in a curved space, there is no way of saying when you have two vector at two different points on the manifold/in space, "are they the same? Are they parallel?" That is not a well defined notion! Because to compare them you need to move them next to each other, and the best you can do to move it is to parallel transport it, and that is going to depend on the path you take to parallel transport it. In general relativity this shows up via the fact that there is no absolute, well defined velocity between two things at different points in space, because there is no way of comparing them. You can likely guess what we want to do next - we want to shrink everything that we have done so far to an infinitesimal size. So here is what we end up doing in Riemannian geometry to characterize the curvature at every point in space: * we define a *infinitesimal* **loop** around which we will parallel transport a vector * But wait, wouldn't a loop require an infinite amount of information to tell you every point on the loop? * We can actually do it with just two vectors! Below we just define $v_1, v_2$ and then move in the direction shown by the neon yellow path. So with just vectors we have defined our "loop"  * Along the way we can parallel transport a vector, $v_3$, and see what happens to it.  * The payoff is that this will then actually define a fourth vector, $v_4$, that tells us how much the original vector changed when going around this loop. We do this with infinitesimals at every point in space  * At the end of all this we get a map! It is a map from three vectors to one vector: $M(v_1, v_2, v_3) \rightarrow v_4$ * This map $M$ is like a little machine that says if you give me three vectors I will spit out a fourth vector. The is known as the **Riemann Curvature Tensor**. A [tensors](Tensors.md) is a generalization of a vector. A vector is a set of components (with respect to some basis). A tensor is a bigger collection arranged in columns, rows, matrices, etc. But its purpose is to map one set of vectors to another set of vectors. So, the RCT specifies at every point, how much does a vector parallel transported around a loop move? This is a lot of information! Because you need to get the value of $v_4$ for every possible $v_1, v_2, v_3$. Now thankfully because everything is vectors we only need to know how this works for the basis vectors and then we can figure out the rest. The RCT is a tensor that describes the total amount of curvature at every point in space. ### Taking a step back If we take a step back for a moment, what we really did here for definition (1) of a straight line is say: * You need to give me some data. You need to give me some information about the manifold whose curvature you want to know. I can then figure out what the curvature is on the basis of that data. * What is the data in this way of thinking? It is the *metric*. It is the way of calculating the distance along a little tiny infinitesimal path, that we can then add up to get the distance along any path. From that we can figure out what a straight line means. From that we can calculate areas, volumes, etc. * The **metric tensor field** is the fundamental thing in general relativity But we also had another way of talking about a straight line in (2), keep walking in a straight line and keep things parallel transported: * Notice that the idea of *distance* did not appear anywhere. We didn't need to know what the distance was or how to calculate it. How is that possible if the whole foundation of geometry was this metric thing that told me the distance, how do we get another way of calculating the curvature without even mentioning the distance? * The answer is that we needed to be given some data once again. This time that data was "what does it mean to parallel transport vectors?" How do we know how to keep a vector constant as we move it along a path. * Parallel transport is a way of comparing vectors at different points in space. It is a way of connecting what is going on at one point in space to another. It requires data/information, answering the question "what does it mean to keep things parallel?" in the form of a **connection** which is a field. Just like the metric which tells me what is the distance along an infinitesimal path (it exists at every point, it is a *field* that can take on different values), the connection is the answer to the question "how do I parallel transport vectors?" and it is also a field. * So at every point I have a way of parallel transporting vectors in every direction. It is a complicated mathematical object and we call it a *connection*. > A **connection** is a *field* that tells you how to parallel transport things. We can then summarize as follows: > The **metric** defines the **connection**. And the connection defines the **curvature**. ### Comparison Recall the following comparison: 1. **Newtonian/Laplacian Physics** - start with a point, chug forward via $F = ma$ 2. **[Principle of Least Action](Principle%20of%20Least%20Action.md)** - take the whole path and minimize the action along the path You get the same answers using *either of the above approaches*; they are equivalent. You may think: is this analogous to the two ways of defining straight lines: 1. Whole path and find minimum length 2. Parallel transport direction / momentum vector Yes! They are completely analogous! It is the differential vs integral version approach. --- Date: 20240613 Links to: Tags: References: * [The Biggest Ideas in the Universe | 13. Geometry and Topology - YouTube](https://www.youtube.com/watch?v=kp1k90zNVLc&list=PLrxfgDEc2NxZJcWcrxH3jyjUUrJlnoyzX&index=26&t=2717s)