Parallel transport Definition and 70 Threads

Homework Statement If I have a two curves \gamma_{1}, \gamma_{2} with the same start and end points, lying on a smooth manifold M. For a vector v at the "start" point, if I parallelly transport down both curves to the "end" point, will the two vectors at the "end" be different or the same...

If I know the metric everywhere, and I specify a closed path, how can I calculate whether a vector parallel transported around the path will return to being the same vector or not? I assume there is some simple integral to describe this, but I'm not sure how to write it down. Unfortunately...

I'm trying to understand parallel transport and I'm stuck. The example given is if you have a unit sphere and you take one of the latitudes (not the equator), take at a point on the latitude the tangent vector to the curve, and parallel transport it around the curve. I don't understand why the...

A vector field Y is parallely propagated (with respect to the Levi-Civita connection) along an affinely parameterized geodesic with tangent vector X in a Riemannian manifold. Show that the magnitudes of the vectors X, Y and the angle between them are constant along the geodesic.

Hi! I've finally decided to tackle a diff geom book, but I'm having trouble with this Problem 4/Chapter 2 from Do Carmo's Riemannian Geometry: Let M^2\subset R^3 be a surface in R^3 with induced Riemannian metric. Let c:I\rightarrow M be a differentiable curve on M and let V be a vector...

This is what I currently understand by parallel transport. The definitions I've read don't talk about it in quite this way, as a vector field, but I think this is equivalent to them: Given a tangent vector V0 at some point P0, construct a vector field along an oriented curve P(?), where ...

Could anyone give me a descriptive picture on WHY geodetic precession occurs? I understand the equations from which it follows, so I can derive it algebraically, but I would like to get an intuitive feeling of why it occurs too. My problem is the following: parallel transport of vectors along...

I'm trying to show that \frac{d}{dt}\; g_{\mu \nu} u^{\mu} v^{\nu} = 0 in the context of parallel transport (or maybe not zero), and I'm rather insecure about the procedure. This is akin to problem 3.14 in Hobson's et al. book (General Relativity an introduction for physicists). As a guess, I...

Hi all, Sorry if this is a dumb question, but what exactly do we mean by the term parallel transport? Is it just the physicist's way of saying isometry? Also, in my class we have just defined geodesics, and we're told that having a geodesic curve cis equivalent to demanding that the unit...

A vector field is parallel transported along a curve if and only if the the corariant derivative of the vector field along the path is 0. That is \frac{d}{d\lambda} V^\mu + \Gamma^\mu_{\sigma \rho} \frac{dx^\sigma}{d\lambda} V^\rho = 0 This is basically what every book says. But what...

In a Stokes theorem, the integral of all curls of a vector field enclosed in some region is equal to the line integral around the boundary. I'm wondering if a similar theorem exists for parallel transport. The Riemann curvature tensor gives a change in a vector when parallel transported...

I'm a bit confused about parallel transport. We demand that the absolute covariant derivative of our (generalised) coordinates is zero along some curve in some frame. Does this really make our vector stay parallel? What if these coordinates were angles or some other general curvilinear...

Parallel transport, as one means of quantifying the curvature of a coordinate space, enables changes in a vector's components, when it is carried around variously oriented loops in that space, to be properly measured, i.e. by comparisons made at the same location. Those changes which are...

How do I obtain the Levi-Civita connection from the concept of parallel transport? So Do Carmo asks to prove that for vector fields X, Y on M, and for c(t) an integral curve of X, i.e. c(t_0) = p and X(c(t)) = dc/dt, the covariant derivative of Y along X is the derivative of the parallel...

Homework Statement Consider a closed curve on a sphere. A tangent vector is parallel transported around the curve. Show that the vector is rotated by an angle which is proportional to the solid angle subtended by the area enclosed in the curve. The Attempt at a Solution First, I parametrize...

If we have as a manifold euclidian R^2 but expressed in polar coordinates... Do any circle centered at the origin constitute a geodesic? Because I think it parallel transport its own tangent vector.

The parallel transport equation is \frac{d\lambda^{\mu}}{d\tau} = -\Gamma^{\mu}_{\sigma\rho} \frac{dx^{\sigma}}{d\tau} \lambda^{\rho} If I take the derivative of this with respect to tau, and get \frac{d^2\lambda^{\mu}}{d\tau^2} =...

I'm reading about bundes and connections but I cannot get past a little problem involving path-ordered exponentials. I hope someone can help me out. I'll try to state the problem as well as possible with plain text LaTeX. My question is just this: How does the the integral (of a general...

hi, i am trying to show that the amount by which a vector is rotated by parallel transport around a triangle whose sides are arcs of great circles equals the excess of the sum of the angles over 180 degrees. this is what i have found out so far call the angles of the triangle (assuming...

I learned that there exists a difference between Lie transport and parallel transport and what that difference is in differential geometry, but I'm getting all confused again when I read the explanation given in the 'intro to differential forms' thread (below). (posted by jeff) How do we do...