I will discuss these notes on Differential Geometry.
Why study vector fields, differential forms etc on a manifold, when you can directly study the whole manifold itself? Well, what does studying the whole manifold mean? One might say “Just look at it! You’ll see the shape and other geometric features”. That is true. However, what the manifold “looks like” depends on how it has been embedded in the space in which you are an observer. The embedding may change the position of the manifold with respect to you, its size, etc. Also, “looking at” a manifold might be difficult in practice. If you are a human living on Earth (which I am assuming everyone reading this blog post is), how do you look at the whole earth without taking a spaceship out of the planet? Can you still study some geometric or topological properties of the planet without shooting into space and looking down on it? Another related question is how do we study the shape and geometric properties of the universe that are part of? We clearly can’t get out of it just yet. Is there still a way to say something meaningful about it? These questions can be answered by studying things like geodesics (paths taken by light), etc. It is study that we wish to undertake when studying Differential Geometry. Although Einstein mainly focused on local properties of geodesics (how two parallel geodesics diverged in a small neighborhood of a point), we can also study global properties like the number of zeroes of a vector field, etc.
Why study tensors? Aren’t vectors and co-vectors enough to study a manifold? Clearly they’re not, as we don’t yet have a notion of “size” yet, which is imparted to the manifold by a metric. Then comes the notion of a curvature, which is yet another tensor which quantifies how far the manifold if from being “flat”. Hence, just considering a manifold with local coordinate systems with vectors and co-vectors is not really enough. Very often we need to impose further information like metric, etc in order to study the manifolds that we encounter in the real world.
Why is it important to parallel transport anything? Well imagine the path of a ball. If its velocity vector is being parallel transported, then it is not under a non-zero net force, telling us that we are in an inertial frame of reference. However, if its velocity vector is not being parallel-transported, then we know that the ball is under a non-zero net force. Hence, parallel transportation tells us something fundamental about the systems under study.
Why do we want to work with coordinates at all, if we already have a metric? Because vector fields, and other tensor fields are often expressed locally in terms of coordinates. Hence, it is often useful to also be able to work in coordinates. But why do we care about the coordinates induced by the exponential map in particular? One reason is that we all Christoffel symbols are (although their derivatives might not be). This makes calculations easier. But couldn’t we have created another coordinate chart where the Christoffel symbol would also be ? Doing so by solving a system of differential equations gives us an overdetermined system. Hence, it is not clear that other such charts exist. But why did we not see the exponential map before? Why only in the case of Riemannian manifolds? There is nothing special about the exponential map if you remove the properties of the metric. It is just a map to an open subset of Euclidean space. What makes the exponential map important in Riemannian geometry is that the straight lines in Euclidean space are mapped to geodesics on the manifold. Hence, this is not an ordinary coordinate chart. This preserves important properties of the metric. However, it is not an isometry because the manifold might have non-zero curvature while Euclidean space does not.
What does two manifolds being isometric means? Does it mean that they have the same metrics in any coordinate chart? No. It just means that one manifold can be mapped onto the other homomorphically such that the pullback metric is the same as the original metric. On an intuitive level, I should be able to map one manifold onto the other without changing lengths, derivatives of the metric, etc. When we take a flat manifold and map it homomorphically to a manifold with non-zero curvature, we are changing the metric. Hence, the map is no longer isometric.
Why do we use these fancy exponential functions to construct bump functions? We can construct an infinite number of differentiable functions that are in some domain and in another. However, these bump functions are also smooth, and it is difficult to construct smooth functions with these properties. That is why we use these fancy functions. Why do we want partition functions anyway? This is so we can use local coordinates to do our calculations. An analogy is that if we want to measure the weight of a heavy object with a small weighing machine, we can break it up into smaller parts, weigh each of those parts individually, and then add those weights up.
What does a differential structure even mean? We’d only heard of differentiable functions before this. Well, a differentiable structure is a bunch of coordinate charts with differentiable transition functions. But who cares about transition functions? What about the interiors of those charts? The interiors look like Euclidean sets. Hence, they’re “smooth” anyway. Why do we care for transition functions anyway? When we’re switching coordinate charts, the representation of a function also changes. But if a function is smooth in one coordinate chart, we want it to remain smooth in another. Hence, the transition functions need to be smooth as well.
Why is it important to make the tangent space closed under Lie bracket multiplication? One answer is that we can now calculate things like the number of generators, which might help us in classifying manifolds. Another way of seeing it is that is the Lie derivative of one vector field with respect to another. Hence, we want the set of vector fields to main closed under differentiation, much in the same that that the set of functions (all differential forms, in fact) remains closed under differentiation.
What does the Jacobi identity mean? It just means that the Lie bracket is a homomorphism. Why is it important for us to have homomorphisms? It ensures that the image also has the same algebraic structure as the domain. But why is that important? Isn’t it too strict a condition? This answer says that homomorphisms preserve algebraic structure, and that is good. But why is it good? One of the main purposes of homomorphisms is classification. We want to tell two algebraic structures apart. One way to do that is to ask in how many ways can I map one algebraic structure on to the other such that the algebraic structure of the former is preserved? If there are no ways, then the two structures are different. Hence, a homomorphism facilitates in comparing algebraic structures. The Jacobi identity facilitates in making a structure-preserving automorphism. This is not true for group actions in general. Why was it important to do it in this instance? Maybe mathematicians saw that this was true for vector fields, and wanted to impose it on all Lie algebras. I’m not sure.
What does “covariant” mean here? When coordinates change, some mathematical objects have an easier law of transformation, whilst others have a much more difficult law. For instance, vectors need just be multiplied with the Jacobian matrix, and they’re good to go. However, second derivatives of functions have a much more complicated law of transformation, for instance. The notion of derivatives given above has a “covariant” law of transformation, and that is why it is called a covariant derivative.
Let us actually take a small detour to study the wikipedia article on covariant derivatives.
So Christoffer symbols were first used to denote curvature, and not differentiation? That would make sense. People were looking for ways to measure curvature. They already knew how to differentiate. It is changing our conceptions of the more “elementary” notions that takes more time.
“Covariant” means independent of the coordinate system. How is that different from invariant? Well, invariant means absolutely unchanging in any coordinate system. For example, the number remains the same number regardless of whether I am looking at it or you are. However, the velocity of a ball does indeed change depending upon the observer. It might be moving to your left, for instance, and to my right. “Covariant” implies that the law of transformation is “simple”, and involves just a multiplication with a matrix. Although it isn’t “invariant”, it is the next best thing. Things don’t change too wildly. How can we have the same expression for two different coordinate systems? How do we know that the two different coordinate expressions are “really” the same? The fact that one can be converted into another isn’t enough of a reason. Simplicity has nothing to do with anything. Well, they give us the same observables. If we were to measure the length, acceleration of any other quantity based on those two different coordinate expressions, they would be exactly the same. An analogy is that although there is no clear way of saying that the moon that I’m watching in the sky is the same as the moon that you’re watching, all the properties of the moon that we can observe like the color, size, etc will be the same. Hence, that is one way of proving that we’re indeed looking at the same moon.
How do regular coordinate derivatives change? When we use the right transformation matrices, everything transforms smoothly (although perhaps not linearly). However, there is no invariant way to write a derivative. For instance, does not denote the same object in different coordinate systems. Is that all that this has been about? Being able to write an object without referring to coordinates? Why can’t we just write , without referring to any coordinate system? Well, we want to be able to write down the formal expression in terms of coordinates, without writing down which coordinate system we are referring to. But why is that important? Writing down a quantity in coordinates is akin to writing down a quantity for an observer. Having something in formal coordinates but without reference to a particular coordinate system implies that we can now write down physical laws (which have to be written in coordinates because physical laws correspond to observers and reference frames) without singling out any coordinate system. This is what it has all been about. Writing physical laws.
Why are we concerned with transformation laws at all? Who cares how objects transform when changing coordinate systems? Well a vector transforms a certain way, and a co-vector transforms another way. Then come tensors, which transform in completely different ways. How do we classify them? Well indices transform like indices transform like vectors, and indices transform like co-vectors. Hence, we may think of them as the tensor product of vectors and co-vectors. Hence, we are able to classify mathematical objects based on how they transform. This becomes even more pronounced in Physics, when certain tensor products of spinors transform as vectors, and that is why we treat them as vectors. But, still, who cares how objects transform? Don’t we care what objects actually “are”? But what really “is” an object, like say a vector? Should it feel hot to the touch, or perhaps denote velocity or acceleration or something of that kind? No. It is just a mathematical object that satisfies certain rules like the transformation laws. Hence, we can classify all objects that satisfy all of those rules and laws as vectors.
How is “covariant” different from “invariant coordinate representation”? It’s very different. “Covariant” just refers to the fact that we have a vector here, and not a co-vector.
The “coordinate grid expands and twists” portion explains why we have Christoffel symbols. But expands and twists with respect to what? With respect to Euclidean space. We have taken Euclidean space to be the epitome of “flatness”, and any coordinate system that twists and turns with respect to it can be said to have non-zero Christoffel symbols. This is not just arbitrary. Twisting and turning with respect to the Euclidean space can be measured in the form of a non-inertial force in Newtonian mechanics. Of course things are slightly more complicated in General Relativity.
What does the covariant derivative have to do with parallel translation? Parallel translation just means that things are moving in a “straight line”, as they should. Moreover, they’re moving in a straight line in all possible coordinate systems. Hence, this is a true fact about the universe, and is not observer dependent.
Let us now get back to the notes.
Why do we need to induce a connection on the submanifold at all? Well, the Earth is a submanifold of the universe. We are only concerned with the velocities and accelerations of balls with respect to the connection induced on the Earth, and not the overall velocity or acceleration vectors of the ball. Hence, there is value to be found in inducing a connection on submanifolds.
Why are transformation laws written like , etc? That is a succinct way to represent a column in the Jacobian matrix. I am surprised that Christoffel symbols don’t transform like tensors though. I only now realize that Christoffel symbols are not tensors.
This is all for today. I hope to upload notes from the next few chapters later in the week.