Thank you so much for this video! It is very clearly explained and you helped me out a lot so far with understanding differential geometry. A question I have is the following and I hope you might want to answer it if possible: Is there some kind of relation between what is going on with these kinds of covariant derivatives and what is happening when we express a tangent vector in a Lie algebra when we are talking about Lie groups? Because we also have some tangent vector on a manifold in a Lie group and we can express this tangent vector in the Lie algebra.
Thank you for that. Yes, these results apply for those manifolds that possess a group structure. For more information on this you might like to look at the following paper; W. Graeub, Lieshe Grupen und affin zusammenhangende Mannigfaltigkeiten, Acta Math. (1961) pp. 65-111. Also, there are books on Lie groups and differential geometry.
I have a few issues here. You say that the tangent bundle is the set of all tangent spaces. I think that is technically incorrect. That is contradictory to being the set of all tangent vectors. It is the set of all _elements_ of tangent spaces. No tangent space is an element of the bundle, only the vectors are. And being disjoint allows them to be labeled according to which tangent space they belong to. And I think this is important in the definition of the manifold, because it tells us which tangent vectors are in an open set. When I read about this all I found was that the tangent bundle comes with a "natural" topology, but not how that is defined... But isn't the job of the connection explicitly dependent on how the open sets of the manifold and the bundle map to each other? Isn't that how we define "nearby" here? That is something I have not been able to properly understand. How does the connection actually connect "nearby" tangent elements. So it seems _critically_ important that the bundle is a manifold(has a topology). Whenever I see people explain parallel transport, they inevitably at some point say "keep the vector parallel to itself". *What does that mean?!* Isn't that exactly what the connection defines? Keeping the vector parallel to itself requires comparison of along the curve. But that is what we are trying to define in the first place. So it is important to be clear that the connection is not unique. It is *some* definition of how to compare whatever "nearby" tangent spaces mean. Can you recommend a resource where I can find more details about the process of actually defining a connection?
In relation to the concept of a Tangent Bundle have a look at this link before reading the response below: www.quora.com/What-is-an-example-of-a-tangent-bundle 1. Tangent Bundle: - The tangent bundle of a manifold is a construction that allows us to associate a tangent space to each point on the manifold in a smooth and consistent manner. - Mathematically, if M is a smooth manifold, the tangent bundle TM is the disjoint union of all tangent spaces T_pM at each point p in M . - Each element of the tangent bundle is a pair (p, v), where p is a point in M and v is a tangent vector in the tangent space T_pM at that point. - The tangent bundle itself is also a manifold, and it has a natural topology induced by the smooth structure of the manifold M . 2. Affine Connection: - An affine connection on a smooth manifold is a way to compare tangent vectors at different points in a smooth and consistent manner. - It provides a rule for differentiating vector fields along curves on the manifold. - Geometrically, an affine connection tells us how to parallel transport tangent vectors along curves on the manifold. It essentially defines what it means for vectors to be "parallel" along a curve. - Mathematically, an affine connection is a choice of a covariant derivative operator that satisfies certain properties, such as linearity and compatibility with the smooth structure of the manifold. 3. Parallel Transport: - Parallel transport is a process of moving a tangent vector along a curve on a manifold while keeping it parallel to itself. - In the context of an affine connection, parallel transport is achieved by transporting a vector along a curve such that its directional derivative along the curve, as measured by the connection, is zero. - Parallel transport is essential in understanding geometric properties of manifolds, such as curvature, and plays a crucial role in various areas of physics and mathematics, such as general relativity and differential geometry. 4. "Natural" Topology: - The "natural" topology on the tangent bundle is typically induced by the smooth structure of the base manifold. - This means that the tangent bundle inherits its topology from the manifold itself, often via the notion of smooth maps between manifolds. - Specifically, given a smooth manifold M , the tangent bundle TM inherits its topology from the product topology on M x R^n , where n is the dimension of M. - This topology allows us to define notions of continuity, convergence, and differentiability on the tangent bundle in a way that is compatible with the smooth structure of the manifold. As for resources to learn more about these concepts and the process of defining a connection, textbooks on differential geometry and Riemannian geometry would be valuable. Some recommendations include: - "Differential Geometry of Curves and Surfaces" by Manfredo P. do Carmo. - "Riemannian Geometry" by Manfredo P. do Carmo. - "Introduction to Smooth Manifolds" by John M. Lee. - "Riemannian Geometry and Geometric Analysis" by Jürgen Jost. These texts provide comprehensive treatments of differential geometry and affine connections, offering insights into the theory and applications of these concepts.
@@TensorCalculusRobertDavie Thank you very much. I am currently working through Carroll's "Spacetime and Geometry" and he is, at points, very cavalier about certain underlying details... Or maybe I lack prerequisites he assumes. So when I try to physically do parallel transport, say by carrying a stick around a closed path, am I implicitly choosing a connection by deciding what it means to keep it parallel to itself? Is the angle I hold my arms at a valid connection? The example of the ant on the surface of a sphere is often given, but how does the ant know what parallel means? Or does this process assume a local metric and automatically results in the Levi-Civita connection?
@@TensorCalculusRobertDavie "1: The tangent bundle itself is also a manifold, and it has a natural topology induced by the smooth structure of the manifold M ." ... "4. ...: inherit ' ... " IOW: The Manifold M has a topology (by definition) and each tangent space has a topology (as it's a vector space and we can use the "epsilon ball" topology introduced by the R^n being the coordinates when defining some set of basis vectors) so the "natural" topology needs to be constructed so that a projection as well to the "point" part as to the "Tangent vector" part reproduces the appropriate open sets.
@@narfwhals7843 I feel that the algorithm the ant uses to decide what is parallel is a "built in" feature of the ant, i.e. an "arbitrary" feature of the "world" it sees and it's evolution. Now the ant can do parallel transport and from this try to construct a Riemann Curvature Tensor field for it's surface. Hopefully it finds that same is clearly defined (not depending on the rotation direction of the closed path etc: metric compatibility and rotation-free). So it seems the Definition of the "parallel" Algorithm is the precaution of the Algorithm to find a Riemann Tensor field. And later it can use same to define a compatible metric tensor field for the "surface" Manifold.) (I.e.: it does not make sense to ask how the ant gets the feeling of parallel, unless there already is a metric on the manifold, which in fact would make the process circular.)
Hello Tarun and thank you for your question. The tilda symbol "~" is only used to represent a vector. Vectors belong or "live" in the tangent space to a manifold.
I have absolutely no idea why lecture notes goes through all the extra bs with covariant derivative when it's simply add differentiating the tangent vector.
