194 Spring 2025 - Reading Course
194 Spring 2025 - Reading Course
Meeting Times: 2PM Fridays, Skye 268
Smooth Manifolds (SM) - Lee
Riemannian Geometry - Lee
Topics - skipped due to admin delays
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Topics - Topology, topological manifolds
Notes: the most important property of topological manifolds is the locally Euclidean property, which tells us if we zoom into a manifold enough, it should look like Euclidean space. You should check that the existence of a local homeomorphism to R^n, n-ball, and an open subset of R^n are all equivalent (exercise 1.1 in SM).
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Topics - Multi-linear algebra, smooth manifolds
Notes:
Given two vector spaces V and W of dimension m and n with bases e_i and f_j, one can construct the direct sum vector space using elements of the form (e_i, 0), (0,f_j) as a basis. This new vector space has dimension m+n.
Instead of the direct sum, one can construct a tensor product from V and W which has dimension mn. The construction is as follows, first take elements of the form (e_i,f_j) as the basis, and then mod out the "multilinear relations": (v_1+v_2, w) - (v_1, w)-(v_2, w), (v, w_1+w_2) - (v, w_1)-(v, w_2), (av, w) - a(v, w), (v, bw) - b(v, w).
Fact: there is a universal property of tensor products. If you have a multilinear map L from V_1 x .... x V_k to R say, then it induces a linear map on the tensor product formed by V_1, ... V_k.
Remark: The collection of multilinear maps on V_1 x .... x V_k is exactly the tensor product of the their dual space. See here.
Remark 2: Our primary vector space of interest is the tangent space at a point of the manifold and its dual, the cotangent space.
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Topics - Tangent Space, Tensor Fields
Notes:
There is a vector space isomorphism between derivations of smooth functions at a point on M and the equivalence classes of curves on M going through the point with the same velocity vector.
We will mostly think about tangent spaces as a span of derivations.
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Topics - Connections, Parallel Transport
Notes:
Parallel transport of a tensor along a curve preserves length and directions, but what is a "constant direction" if the tangents spaces vary from point to point?
One way to make sense of this is introducing a unique connection based on a prescribed Riemannian metric.
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Topics - Riemannian Metric, Curvature Tensor
Notes:
A (Riemannian) metric is a symmetric smooth (0,2)-tensor that is positive definite.
The non-degeneracy of the metric allows one to move back and forth between vectors and covectors (cf. Riesz reprensentation).
The (0,4)-tensor R_{abcd} was introduced.
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Topics - Riemannian Manifolds, Integration, Time Orientation
Notes:
We spent some time discussing previous sections.
The Levi-Civita connection is uniquely determined by the metric so carries geometric data of the manifold.
This addressed one of the questions in a previous week regarding animations on the Wikipedia page of parallel transport - clearly both pictures preserve something, but that something is determined by the metric.
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Topics - Einstein Equations and Schwarzschild Solution
Notes:
We discussed actions from classical mechanics a little bit. The main take away here is that the critical points of an action functional (an integral) gives us equations of motion. In GR, vacuum Einstein solutions are critical points of the Einstein Hilbert action.
The Schwarzschild metric is a spherically symmetric solution to Einstein's vacuum equations.
The main feature of Schwarzschild for our discussion is the asymptotic flatness at spatial infinity.
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Topics - The Cauchy Problem in General Relativity & Dominant Energy Condition
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Topics - The Positive Mass Theorem
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