What is the Equation for Finding the Minimum Area of a Non-Euclidean Surface?

[eljose]

If we have that for geodesic they satisfy...

$$S=\int_{a}^{b}ds\sum_{a,b}(\dot{x_{a}\dot{x^{b})$$

then minimizing the functional we get the geodesic equation..my question is if for the Area of a Surface is:

$$A=\iint_{S}dA\sum_{a,b}(f_{a})(f_{b})$$

wherewehave defined $$f_{a}=df/dx_{a}$$

then my questions are..

-you can see that thes definitions can be generalized to the case we have a non-euclidean metric g_ab, What would be the equation that gets the minimum of the surface?...

-What is the "Geodesic surface" (MInimal surface) equation depending on the connections and the metric?..does is satisfy that $$\nabla{f}$$ where f is the surface and "nabla" is the COvariant derivative.

-What is the equivalent of the $$R_{ab}$$ tensor for our minimal surface?..thanks...note the minimizes J or J^{1/2} you get almost the same results in the practise.

 

[Doodle Bob]

OK, that first sentence makes absolutely no sense both grammatically and mathematically.

there are a few definitions you need to be aware of. (M,g) is a Riemannian manifold with covariant derivative $$\nabla$$.

1. The length of any curve $\gamma:[a,b]\rightarrow M$ is given by
$l(\gamma)=\int_a^b g(\gamma_*(t),\gamma_*(t))dt$, which is the S that you have above.

2. A geodesic of a Riemannian manifold is a curve $\gamma$ whose tangent vector is parallel to itself, i.e. $\nabla_{\gamma_*}\gamma_* =0$.

It is not necessarily guaranteed that a geodesic minimizes length. On the sphere in 3-space, geodesics are portions of the great circles. In particular the great circle that starts at the north pole and wraps all the way back to the north pole is a geodesic that clearly does not minimize lengths.

In fact, it is not guaranteed that any two points are even connected by a geodesic, e.g. the plane without the origin.

You have misstated the result, which is: given two points a and b on a Riemannian manifold, if there is a curve connecting a and b the length of which is the least of all curves connecting a and b, then that curve is a geodesic.

The analogous result for surfaces in 3-space is Plateau's Problem. The key tensor here is the mean curvature of the surface. In this case, you need to be given a curve in 3-space that is the boundary of the surface. I suggest you look this up in Do Carmo's text on curves and surfaces.

 

[eljose]

umm. well what i want is the equivalent to the problem of minimizing the area of a surface $$\delta{A}=0$$, the purpose is to give some Geometrical meaning to Quantum mechanics in a similar way Einstein did with G.R from the Physical point of view:

-The wave function $$\Psi(x,y,z,t)$$ is "somehow" a Minimal surface in R^4 so if we have a covariant derivative or similar the Schroedinguer equation can be written as$$\nabla{\Psi}=0$$ (1) where this last equation comes from the "differential equations for a minimum surface"

-There exist an equivalent to Riemann Tensor R_ab in R^4 so its 00 component is precisely Schroedinguer equation,the "boundary" of this curve would be $$\Psi(x,y,z,0)$$ which can be perfectly defined.

 

[Timbuqtu]

I don't really get what you're saying, but have a look at the Nambu-Goto action in bosonic string theory. It is just given by the area of the (two-dimensional timelike) worldsheet swept out by the (one-dimensional) bosonic string. So in this case variation of the action/area gives us the classical equations of motion for the bosonic string. See for instance

http://en.wikipedia.org/wiki/Nambu-Goto_action"

 

[eljose]

My main purpose is reformulate QM in a geometric way similar to what Einstein did so somehow the Wave function is a Geodesic surface in the sense that: $$\nabla_{\Psi}{\Psi}=0$$ (paralell transpor of the surface) then the general way to define the Quantum dynamical of the wave would be $$K_{ab}=T_{ab}$$ where K plays the same role of Riemann tensor R_ab and T_ab is the energy-momentum tensor

 

[Doodle Bob]

My main purpose is reformulate QM in a geometric way similar to what Einstein did so somehow the Wave function is a Geodesic surface in the sense that: $$\nabla_{\Psi}{\Psi}=0$$ (paralell transpor of the surface) then the general way to define the Quantum dynamical of the wave would be $$K_{ab}=T_{ab}$$ where K plays the same role of Riemann tensor R_ab and T_ab is the energy-momentum tensor

Wow, that's quite so programme to embark. Isn't this essentially what Penrose has been trying to do for the past 50-odd years?

The transistion from curves to surfaces, I'm afraid, is not nearly as analogous as one would at first expect. Area-minimizing surfaces (given a boundary) and totally geodesic submanifolds are actually two different classes of submanifolds. There's often an intersection between the classes but there are plenty of minimal surfaces that are not totally geodesic surfaces. Plus there are totally umbilical submanifolds...


As I've stated before, it would seem that you've got some homework to do before embarking on this. A nice review of Riemannian geometry can be found in "Structures on Manifolds" by Yano and Kon (World Scientific Press, 1984). It helped me get through grad school.


I'm not very familiar with wavefunctions but this equation $$\nabla_{\Psi}{\Psi}=0$$ doesn't seem to make much sense. The object in the subscript needs to be a vector, not a function, so I'm guessing that you mean you're looking for a submanifold on which Psi is parallel, i.e. A $$N\subset M$$ on which $$\nabla_X{\Psi}=0$$ for all $X\in TN$.

 

[eljose]

and appart from Riemann Tensor $$R_{ab}$$ what other curvature tensors we have in Differential Geometry?..i mean if for a minimal surface is easier to deal with some other curvature tensor different from the Riemann one or if only we can define curvature by using Riemann tensor and Riemann scalar $$g^{ab}R_{ab}$$ (we assume an implicit sum over the "a" and "b" -->Einstein notation...)

 

[Dcase]

Ivars Peterson has an interesting and appropriate minimal surface article for this thread in Science News 17 Dec 2005; Vol 168, No 25 , p 393 entitled 'Surface Story: Inspired by spiral soap films, mathematicians zero in on a novel, economical, and infinite helix'.

This discusses the Weber, Hoffman and Wolf proof of a genus one helicoid [with a handle] from 15 Nov 2005 Proceedings of the National Academy of Sciences.

This pertains to “an infinite minimal surface called a helicoid” which I think has also been described as a space dividing curve. “Twisting the ordinary two-dimensional plane into a helicoid converts the plane's flatness into saddle-based curviness.”
http://www.sciencenews.org/articles/20051217/bob9.asp

This may be an approach to relate Penrose spinor theory to Penrose twistor theory or perhaps even loops to helical strings as in 'stroops'?

 

[coalquay404]

and appart from Riemann Tensor $$R_{ab}$$ what other curvature tensors we have in Differential Geometry?..i mean if for a minimal surface is easier to deal with some other curvature tensor different from the Riemann one or if only we can define curvature by using Riemann tensor and Riemann scalar $$g^{ab}R_{ab}$$ (we assume an implicit sum over the "a" and "b" -->Einstein notation...)

There are other measures of curvature we can use, but the Riemann tensor is often the most useful, particularly in physics. The other obvious notion of curvature is the extrinsic curvature or second fundamental form of a surface which is embedded in a higher-dimensional manifold. Technically speaking, it goes something like this: suppose that $$(\mathcal{M},g)$$ is an $$m$$-dimensional (pseudo)-Riemannian manifold and that we have an atlas $$(U_i,\phi_i)$$. We can define an embedding $$\Sigma\hookrightarrow\mathcal{M}$$ of codimension $$2 \le \sigma < m$$ by specifying it as the level set of some scalar function:

$$f(x)=0.$$

At each point $$p\in\Sigma$$ we have a unit normal vector $$\mathbf{n}$$ to the surface. To visualise, suppose that the manifold $$(\mathcal{M},g)$$ is actually a space-time and that $$\Sigma$$ is a space-like hypersurface of codimension one. Then the unit normal vector is timelike, where

$$n^a \equiv\frac{g^{ab}\partial_b f}{\sqrt{-g^{cd}\partial_cf\partial_df}}$$.

You can then define the extrinsic curvature tensor $$K_{ab}$$ by

$$K_{ab} = \perp\nabla_an_b$$

where $$\nabla$$ is some connection on $$\mathcal{M}$$ and $$\perp$$ is a projection operator from $$\mathcal{M}\to\Sigma$$ (you can think of it as the mixed form of the induced metric on the hypersurface).

 

[kaiba]

Hi,


I am not a mathematician nor a physician; am currently working on a project to correlate minimal surfaces and surfaces from deformed rocks formed by geological processes.
I am using meshlab to estimate the mean curvature. Can someone provide me with some simple explanations regarding:-

1. what is the algorithm/formula used to estimate mean curvature?
2. I run meshlab using a sinusoid curve (which is not a minimal surface) but since it essentially consists of planes between the max. and min. point, how come it doesn't show zero mean curvature at this regions? since the plane is a trivial minimal surface?

3. if i have a surface that has both zero and non zero mean curvature, how do i classify it?

if anyone has done any work on these, i hope to hear from you. many thanks,