Take R2. Take a function f(x,y) defined on R2 which maps every point to a real number. The gradient of this at any point mean a vector which points in the direction of steepest incline. The magnitude of the vector is the value of the derivative of the function in that direction. Both of these...
What is the Lie derivative? Is it just like the covariant derivative, but treating a different metric as the flat metric? Is there any way of explaining it that avoids lie derivatives? Or should I try learning lie derivative first. Does the lie derivative require lie algebra to understand?
Thanks for the help as always wannabeNewton. Thanks to you (and others) I've got over my obsession with coordinates at last. I understand congruence's. Null congruence's are just the path that moves through no proper time. And geodesic congruence's are just where at each point in the local...
So say you have a 2D Riemannian manifold with a metric defined on it and for simplicity let's say its flat. That means there exists a coordinate system for which the metric tensor is the normal Euclidean metric everywhere. However let's say we are using an arbitrary coordinate system with a non...
Surely the definition of the geodesic is the path of shortest length between two points which remains in the space?
Derivation - So take an arbitrary coordinate system to describe our D dimensional space, call this X. Then take the Cartesian coordinate system, denote this C. If we have a path...
I understand why the geodesic equation works in flat space. It just basically gives a set of differential equations to solve for a path as a function of a single variable s where the output is the coordinates of whichever parameterization of the space you are using. But the derivation I know and...
In all three cases I know how if you accept that they can be written in that form (I.e. as power series or infinite series of the sines and cosines), then you can derive the coefficients using cleverly picked transformations, i.e. differentiation, the Fourier transforms or Cauchy integral...
WannabeNewton - Could you show me the computation of the Riemann curvature tensor for the paraboloid z=x^2+y^2 at the point (1,1,1). I thought I knew what I was doing but I'm not sure I do. Does parallel transport mean project your vector into the tangent space of the infinitesimally close point...
Is the gauss map just a parameterization of the surface, i.e. each point on the surface is mapped to the point on the unit sphere with the same normal vector as that point on the sphere - or is there more to it than that? Is the second fundamental form then basically just the derivative of the...
Can someone please explain what the second fundamental form is. I know the first fundamental form is the metric, which is used to construct length/dot product of two vectors, so is the second fundamental form the way that you construct the area spanned by the two vectors i.e. cross product? So...
I'm struggling with why covariant differentiation doesn't commute in curved spaces. To define a covariant derivative - you pick a coordinate system in which this new derivative is just going to equal the normal derivative in that coordinate system. Then from that you can define christoffel...
The way I understand the christoffel symbol is as the derivative of the basis vectors. If your in a curved space only certain vectors are allowed at each point (those in the tangent space) ergo my conclusion was that the basis vectors must change and that the christoffel symbol would not be zero?
Another question. Why do we need the Riemann curvature tensor to measure curvature It took me a good few hours just to calculate a couple of components of it for a curved 2d space. It seems as if its just overly complex. Also the christoffel symbols alone could do the same job surely, since if...