Curvature of Manifolds: Understanding Relative Concepts

[stevendaryl]

I'm struggling with why covariant differentiation doesn't commute in curved spaces.

Think about traveling on the surface of the Earth. Suppose you have two sticks joined at right angles. Initially, you're at the equator, with one stick (Stick A) pointing North, and another stick (Stick B) pointing East. Let L be 1/4 the circumference of the Earth. If you travel a distance L in the direction of Stick A, and then a distance L in the direction of Stick B, you will end up distance L to the East of where you started. If instead, you travel in the direction of Stick B, and then in the direction of Stick A, you'll end up at the North Pole, a distance L to the north of where you started. So that it makes it clear that the order of traveling makes a difference.

In contrast, on a flat surface, you can take any two vectors, and if you travel in one direction and then the other, you get to the same spot as if you did it in the opposite order.

 

[Benjam:n]

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 if you say wanted to construct an a surface area

 

[WannabeNewton]

The second fundamental form is given by $II_p(\xi,\eta) = -\left \langle dN_p (\xi),\eta \right \rangle, \xi,\eta\in T_p \Sigma$ where $\Sigma\subseteq \mathbb{R}^{3}$ is a regular surface and $N_p$ is the Gauss map. We call $dN_p$ the Weingarten map; the second fundamental form is what leads to the principal curvatures and principal directions of $\Sigma$ because the second fundamental form is symmetric implying that $dN_p$ is self-adjoint which implies by the spectral theorem that it is orthogonally diagonalize. Its eigenvalues are called the principal curvatures and its eigenvectors the principal directions. The product of the principal curvatures gives the Gaussian curvature and the average gives the mean curvature. One can also compute the Gaussian curvature directly from the coordinate representations of the first and second fundamental forms as $K_p = \frac{\det [II_p]}{\det [I_p]}$. So in a sense, the second fundamental form characterizes the curvature of $\Sigma$.

 

[Benjam:n]

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 normal vectors field on the surface?

Also I understand principle curvatures in terms of osculating circles - so that seems to have a nice link as presumably normal vectors determine the centre of the circle?

Finally is there an equivalent concept to the metric for area, i.e. eijkdyj/dxndyk/dxm. Presumably not because you'd need two of them contracted together to get the modulus and square rooted along with the vectors which they're later going to be contracted with.

 

[WannabeNewton]

Yes to all your questions up till the last. As for the last question, I'm not sure what you're asking (mainly because I can't really read what you wrote down there). Are you asking how we use the natural volume form $\epsilon_{abcd}$ associated with the metric $g_{ab}$ to find volumes?

 

[Benjam:n]

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 and scale it up so it still has the same magnitude. Thanks

 

[George Jones]

Benjam:n, what are you using for references. Are you trying to work systematically through a particular reference? If not, you should be.

 

[stevendaryl]

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 and scale it up so it still has the same magnitude. Thanks

Are you asking for how to do it, conceptually, or how to actually compute the components of the curvature tensor? The latter is an awful lot of tedious work, in my experience, even though there is nothing conceptually hard about it.

The first thing you'd need to do for your particular example is to compute the components of the metric tensor. It's a 2D surface, so the metric tensor has 4 components, which we can call
g_{x x}, g_{x y}, g_{y x}, g_{y y}
(Because the tensor is symmetric, g_{x y} = g_{y x})

To figure out the components, you take two infinitesimally removed points:
(x, y, z) and (x + \delta x, y + \delta y, z + \delta z)
and you compute the square of the distance between them:

\delta s^2 = \delta x^2 + \delta y^2 + \delta z^2

Since z = x^2 + y^2, we can eliminate \delta z via
\delta z = 2 x \delta x + 2 y \delta y

Now write \delta s^2 in terms of \delta x and \delta y, and then you can just read off the components of the metric tensor, because

\delta s^2 = g_{x x} \delta x^2 + g_{x y} \delta x \delta y + g_{y x} \delta y \delta x + g_{y y} \delta y^2

For your example, I think you end up with:

g_{x x} = 1 + 4x^2
g_{x y} = g_{y x} = 2 x y
g_{y y} = 1 + 4 y^2

Next, you need the inverse of the metric tensor, as well. The components are written using raised indices: g^{x x}, g^{x y}, g^{y x}, g^{y y}.

Next, you take derivatives of the metric tensor components to get the connection coefficients, \Gamma^i_{j k} where i, j, k are each either x or y. These are computed in terms of the metric tensor as follows:

\Gamma^i_{j k} = \frac{1}{2} \sum_{l} g^{i l} [\partial_j g_{lk} + \partial_k g_{lj} - \partial_l g_{jk}]

Finally, you take derivatives of \Gamma^i_{j k} and compute the Riemann tensor components using

R^i_{jkl} = \partial_k \Gamma^i_{jl} - \partial_l \Gamma^i_{jk} + \sum_m \Gamma^i_{km} \Gamma^m_{lj} - \sum_m \Gamma^i_{lm} \Gamma^m_{kj}

 

[Mentz114]

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 and scale it up so it still has the same magnitude. Thanks

Does the point (1,1,1) lie on z=x^2+y^2 ?