Lecture 8: Parallel Transport & Curvature (International Winter School on Gravity and Light 2015)

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Recommends user149705 · 2017-02 · Mathematics Stack Exchange · on “Derivation of the geodesic of a smooth manifold”

I have a difficulty in the following derivation of the geodesic equation for a curve. Let $\gamma:R\rightarrow M$ be a curve on the manifold $M$. We have a connection and we are trying to find the geodesic curves for that connection. We define the tangent vector of $\gamma$ in the coordinate system $ \{x^i\} $by : $X_{\gamma} = (x^i \circ \gamma)'\frac{\partial}{\partial x^i} $ \begin{align} \nabla_{X_\gamma} X_\gamma &= 0 \\ &=\nabla_{(x^i \circ \gamma)'\frac{\partial}{\partial x^i}} (x^j \circ \gamma)'\frac{\partial}{\partial x^j} \\ &= \dot{\gamma}^i\frac{\partial \dot{\gamma}^j}{\partial x^i}\frac{\partial}{\partial x^j} + \dot{\gamma}^i\dot{\gamma}^j\Gamma_{ji}^q\frac{\partial}{\partial

Recommends Antoni Parellada · 2020-10 · Mathematics Stack Exchange · on “Connections and metrics on Riemannian manifolds”

He explains at this point [this video] : with a covariant derivative we talk about the **straightest curves**. At this point [LINK] , which allows findings the **shortest curves (geodesics)**, he explains that we can require that the shortest curves are also the straightest curves, and vice versa, so as to get **metric compatible connections,** resulting in connections determined by the metric. In this case, from a metric we get a connection, which yields a curvature; and hence from a metric we get a curvature.

Recommends Clemens Bartholdy · 2022-01 · Mathematics Stack Exchange · on “How do you reconstruct the surface knowing the connection coefficients?”

In this lec-7 on Connections by Frederic Schuller, the connection is introduced as an extension of the directional derivative of scalar fields to $(p,q)$ tensor field. The initial axioms the connection obeys is same as all that which the directional derivative for scalar field obeys. About 35:07 [LINK] , he says that the remaining freedom of how the directional derivative generalizes is determined by the Christoffel symbols At 6:30 of lec-8 [this video] , it is said that the choice of christoffel symbols , fixes our how our manifold looks in space $(M,O , A, \nabla)$ . The point I am confused is at 46:26 [this video] , he concludes the Christoffels of the regular round sphere are given as: $$ \Gamma_{

Recommends tisPrimeTime · 2022-07 · Mathematics Stack Exchange · on “Understanding the implications of the covariant derivative on geometrical shape”

I am self studying differential geometry (and information geometry), and came across the following idea from a lecture by Frederic Schuller: [this video] Essentially, one can define an abstract sphere via, $(S^2, \mathcal{O}, \mathcal{A})$ , where $\mathcal{O}$ is the topological strucutre, and $\mathcal{A}$ the atlas, of abstract sphere structure $S^2$ . From here it is claimed that through one's choice of linear connection (covariant derivative), one can define: The standard round sphere with one choice of covariant derivative, $\nabla$ , so we have structure: $(S^2, \mathcal{O}, \mathcal{A}, \nabla)$ Or an ellipsoidal structure with a slightly different chosen connection $(S^2, \mathcal{O}, \m

Cites Bogorovich · 2016-03 · Mathematics Stack Exchange · on “Is the autoparallel equation same as the geodesic equation?”

My question may sound silly but i am self learning differential geometry using watching these lectures ( Lecture 8 [this video] ) from 2015 by professor Frederic Schuller. Can somebody please tell that the auto-parallel equation same as the geodesic equation?

Cites user123124 · 2019-05 · Mathematics Stack Exchange · on “Connections and metrics on Riemannian manifolds”

In a lecture on connections it was claimed that a manifold "attains a shape" when it is equipped with a connection while there were no mentions of a metric. Is it correct to intuitively think that the connection determines the "shape" via how tangent spaces are "connected" or equivalently a rule how tangent vectors are moved in parallel manner between such spaces, while the metric only determines length of the same tangent vectors(and hence any curve). And therefore the metric structure and connection structure is completely independent in general, i.e two completely separate things, unless your connection is defined in terms of the metric by kind of brute force as in the Levi-Cevita case, w

Cites user211673 · 2022-02 · Mathematics Stack Exchange · on “Proper Way to view vector field component functions with respect to covariant derivative”

I follow this lecture [this video] and at minute 35, the covariant derivative is used to calculate the geodesic equation. The lecturer says, that the term to calculate the second derivative is strictly not defined. He is right, because Gamme dot q, is a function from R to R and not from M to R. So the partial operator is not defined. How does he calculate the second derivative and how to make sense out this. Is it better to think of the vector field component functions as function from M to R? In the lecture they are functions from R to R. Iam grateful for any help!

Cites Clemens Bartholdy · 2022-04 · Mathematics Stack Exchange · on “Concrete example of charting a topological manifold without specifying a connection”

I have been trying to understand manifold theory for a long time now, and every single example of charting a manifold I see is after giving the structure of a connection on it. For instance , consider Visual Differential Geometry chapter-2 in Tristan Needham where he shows the various projections of sphere onto plane, and finally, charting the pseudo sphere. The issue I have with this is, when we say how the manifold actually sits in space, we have already specified a connection. Actually if one thinks about it, it seems impossible to chart a manifold without specifying a connection through geometrical method as without it, we don't have a concrete entity to draw lines and angles on! ( Eg: I

Cites Clemens Bartholdy · 2022-04 · Mathematics Stack Exchange · on “Concrete example of charting a topological manifold without specifying a connection”

6:36 [this video] , he shows how only when he puts on the very specific choice of $\nabla$ does the sphere become the sphere and the ellipse the ellipse as we think it in 3D space (I added a lot of words, but it's wht I understood). Lemme go over what you said again.

Cites Clemens Bartholdy · 2022-05 · Mathematics Stack Exchange · on “Confusion on the intuition of a connection”

At 6:27 of this lecture [this video] by Frederic Schuller, it is remarked that only when we fix the connection (by providing Christoffel's) that we get the specific shape of the sphere that we usually imagine, and, perhaps a choice of a different connection would have ended up in a sphere: [LINK] This is all good and fine. But then I saw this video by Eigen Chris [LINK] , and 17:55 [LINK] of it, he says that one can set all the Christoffels while on the sphere to zero to get another connection named the boring connection: [LINK] Now this is contradictory with the Schuller lecture's to me because of two points: In Schuller's lectures, it is found that the connection fixes the shape, but here Eigen