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I am trying to practice computing the second fundamental form and the mean curvature, and I am trying to compute them for the Pseudo-Sphere in $n+1$ Minkowski spacetime.

Pseudo-Sphere in $n+1$ Minkowski spacetime is defined as: $\mathbb{S}^{1, n}(r):=\left\{x \in \mathbb{R}^{1, n+1} | \eta(x, x)=r^{2}\right\}$. As my first step I found the unit normal as $\nu = \frac{1}{r}x^{i}\partial_i$.

So I try using the definition of the second fundamental form $K_{ij}=g\left(\nabla_{\partial_{i}} \nu, \partial_{j}\right)$, and for the mean curvature its the trace of the second fundamental form.

But I am not sure how to follow through with the calculations. Any help is very appreciated.

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  • $\begingroup$ What have you tried? Given your expression for $K$ is in coordinates, you'll no doubt need to set up coordinate system on $\mathbb{S}^{1,n}(r)$. It might be useful to start with the $n=1$ case for something easier to visualize. $\endgroup$
    – Kajelad
    Jun 15 '20 at 12:38
  • $\begingroup$ @Kajelad I tried to do so but I am having trouble writing down the metric of the pseudo-sphere in n-dimensions in order to use the definition. $\endgroup$
    – Joel
    Jun 15 '20 at 13:12
  • $\begingroup$ What coordinates are you using for the pseudosphere? $\endgroup$
    – Kajelad
    Jun 15 '20 at 15:12
  • $\begingroup$ @Kajelad I am using $\{x^1,...,x^n\}$ in general, not sure if that's what you mean though. $\endgroup$
    – Joel
    Jun 15 '20 at 15:18
  • $\begingroup$ By this do you mean graph coordinates, i.e. the inclusion map has the form $\iota(x^1,\dots,x^n)=\sqrt{r^2+\sum_{i=1}^n(x^i)^2},x^1,\dots,x^n$ ? $\endgroup$
    – Kajelad
    Jun 15 '20 at 15:27
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For clarity, I'll use hats and latic indices $\hat{g}_{ij}$, etc. to denote objects in the submanifold, and no hats and Greek indices $g_{\alpha\beta}$ to indicate objects in the ambient manifold. With the chosen coordinates the inclusion map is a graph: $$ x^0,x^1,\dots,x^{n+1}=f(\hat{x},r),\hat{x}^1,\dots,\hat{x}^{n+1} $$ The objects needed are the coordinate vectors $\hat{\partial}_i$ included into the ambient space (with some extension). These can be computed by chain rule. $$ \hat{\partial}_i=\frac{\partial x^\alpha}{\partial\hat{x}^i}\partial_\alpha=\frac{\partial f}{\partial x^i}\partial_0+\partial_i $$ Given $f(x,r)=\sqrt{\sum_{j=1}^{n+1}(x^j)^2-r^2}$, we can write this out more explicitly $$ \hat{\partial}_i=x^i\left(\sum_{j=1}^{n+1}(x^j)^2-r^2\right)^{-1/2}\partial_0+\partial_i $$ Technically, this is not a proper vector field and is is only defined on the submanifold. However, allowing $r$ to vary as a function of coordinates will provide a suitable extension, as it did for $\nu$.

Once you have these vector fields, as well as $\nu$, all of the computations in the formula $K_{ij}=g\left(\nabla_{\hat{\partial}_i}\nu,\hat{\partial}_j\right)$ take place in the ambient space.

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  • $\begingroup$ The metric indicated in the definition is the one on the ambient space, which is the one that describes the pseudo-sphere(which I still don't know how it looks) unless I am confusing the two. that's why I am still confused about how to carry out the computation. $\endgroup$
    – Joel
    Jun 15 '20 at 16:19
  • $\begingroup$ The pullback metric isn't really needed to compute the second fundamental form, but the computation to push the coordinate vectors from $\mathbb{S}$ to $\mathbb{R}$ applies. I've edited the answer to reflect this. $\endgroup$
    – Kajelad
    Jun 15 '20 at 16:52

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