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I'm reading "Riemannian Geometry" by Manfredo P. do Carmo and I'm trying to calculate the principal curvature of the geodesic sphere of radius $r$ in the space of dimension $n$ and of constant sectional curvature $c$. So far, I can only calculate the case where $c=0$. For the cases where $c<0$ and $c>0$, I really don't know how to start.

Denote by $\overline M$ the space of dimension $n$ and of constant sectional curvature $c$, and by $M$ the geodesic sphere of radius $r$. Then $$\overline M=\left\{ \begin{array}{lll} \mathbb R^n & c=0, \\ \mathbb S^n(c) & c>0, \\ \mathbb H^n(c) & c<0. \end{array} \right.$$ According to "Riemannian Geometry" by Manfredo P. do Carmo, principal curvatures are the eigenvalues of shape operator.

For the case where $c=0$, let's first consider $r=1$. let $p\in M$, $x\in T_pM$ and $\eta\in(T_pM)^\bot$, $|\eta|=1$. Let $N$ be a local extension of $\eta$ normal to $M$. Then \begin{equation*} \begin{aligned} S_\eta(x)&=-\left(\overline\nabla_xN\right)^T\\ &=-\left(\overline\nabla_{X^i\partial_i}N^j\partial_j\right)^T\\ &=-\left(X^i\partial_i(N^j)\partial_j\right)^T\\ &=-\left(X^i\partial_i(N^j)\partial_j-\left\langle X^i\partial_i(N^j)\partial_j , N^k\partial_k\right\rangle N^l\partial_l\right)\\ &=-\left(X^i\delta_i^j\partial_j-(X^i\delta_i^j N^k\delta_{jk})(N^l\partial_l)\right)\\ &=-\left(X^i\partial_i-\left(\sum_i X^i N^i\right)\left(N^l\partial_l\right)\right)\\ &=-X^i\partial_i\\ &=-x. \end{aligned} \end{equation*} Therefore, all the principal curvatures of the sphere of radius $1$ in $\mathbb R^n$ with respect to the outward pointing unit normal $N=N^j\partial_j$ are $-1$. If the radius is $r$, then $N=\frac1r N^j\partial_j$, so the principal curvatures are $-\frac1r$. If the normal direction is inward, then $N=-\frac1r N^j\partial_j$, so the principal curvatures are $\frac1r$. This finishes the case where $c=0$.

For the cases where $c<0$ and $c>0$, the choice of frame $\partial_i$, the choice of unit normal vector field $N$, and the calculation of connection $\overline\nabla$ are quite complicated. Can someone give me a hint or answer? Any help would be appreciated!

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I'll outline two approaches. Both wind up using the function $\phi_{c}(t)$ defined by $$ \phi_c(r) = \begin{cases} \frac{1}{\sqrt{c}} \sin(\sqrt{c} \, r) & c > 0 \\ r & c = 0 \\ \frac{1}{\sqrt{-c}} \sinh(\sqrt{-c} \,r) & c < 0\end{cases} $$

The first approach uses Jacobi fields and abuses the symmetry of derivatives of a variation (Lemma 3.3.4 in do Carmo):

  1. Let $p$ be the center of the geodesic sphere $M$, and let $N$ be the unit normal field pointing outward from $p$. Then $|N| = 1$ implies $(\bar{\nabla}_x N)^T = \bar{\nabla}_x N$.

  2. Let $\gamma$ be a geodesic emanating from $p$ (so that $N = \dot{\gamma}(R)$) and find a Jacobi field $J(t)$ along $\gamma$ such that $J(0) = 0$ and $J(R) = x$. If you write $J(t) = f(t) E(t)$, where $E(t)$ is parallel along $\gamma$, you can compute an explicit expression for $f$ from the curvature condition. Note that $J$ corresponds to the geodesic variation that rotates at $p$ "in the direction of $x$" (as a bit of intuition).

  3. Use the fact that $\bar{\nabla}_J N = \bar{\nabla}_N J$ (Lemma 3.3.4). The expression $\bar{\nabla}_N J$ is easy to compute explicitly from $J(t) = f(t) E(t)$.

  4. Using this approach you should be able to derive $S_\eta(x) = -\frac{\phi_c'(R)}{\phi_c(R)} x$.

The second approach uses the relations for isometric immersions (in particular we'll use do Carmo Theorem 6.2.5) and an explicit formula for the constant curvature metric.

  1. Argue that by symmetry, the shape operator of the geodesic sphere must look like $S_{\eta}(x) = \ell x$ for some unknown constant $\ell$.

  2. Likewise by symmetry, the geodesic sphere of radius $r$ must have some constant positive curvature $\kappa(r)$. (This function depends on the curvature $c$ of the larger space, of course.)

  3. By Gauss's Theorem (6.2.5), conclude $\kappa(R) - c= \ell^2$. This gives a relationship between the (intrinsic, constant) sectional curvature of the geodesic sphere of radius $R$ and the eigenvalues $\ell$ you're looking for.

  4. The constant curvature $c$ metric looks like a warped product $g = dr^2 + \phi_{c}^2(r) h$, where $h$ is the standard metric on the $(n-1)$-sphere. So the curvature $\kappa(R)$ we're looking for is the curvature of a sphere of radius $\phi_c(R)$, which gives $\kappa(R) = 1/\phi_c^2(R)$.

  5. So you derive $\ell^2 = \frac{1}{\phi^2_c(R)} - c$.

Of course, the two approaches give the same result, which follows from trig identites.

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