# Is there any generalization of warp product?

For the simplest example of warp product $S^n=[0, \pi]\times_{\sin t} S^{n-1}$, the geodesic sphere $\partial B(p, t)$ has the highest symmetry, namely the eigenvalue of the shape operator is constant $c(t)$ of multiplicity $(n-1)$.

If we look at the Complex Projective space $\mathbb{CP}^n$ with Fubini-Study metric. The distance sphere has again the same eigenvalues at any point, but this time the eigenvlaues are one with multiplicty $(2n-2)$ and one simple eigenvalue correspond to $J\nabla r$ direction. The metric on $\mathbb{CP}^n$ is not warp product although it can be written in terms of the standard metric on sphere using Hopf fibration. In the homogeneous coordinate of $$\mathbb{CP}^n=\{[z_0, \cdots, z_n]:0\ne (z_0, \cdots, z_n)\in \mathbb C^{n+1}\}$$

the FS-metric on $U_0:=(z_0=1)\cong \mathbb C^n$ can be written as: $$g_{i\bar j}=\frac{\partial ^2}{\partial z_i \partial \bar{z}_{j}} \log (1+|z_1|^2+\cdots + |z_n^2|)$$ This metric is invariant under $SU(n+1)$ action.

I am wondering is there any name for this kind of symmetry? i.e. the geodesic sphere $\partial B(p, r)$ has the same set of eigenvalues counting with multiplicity. The metric should in some sense has the radial symmetry and harmonic manifolds are within this class of metric. Is there any name for this metric?

One more question, is there any description of this kind of Riemannian metric in pure metric geometry terms (i.e. without using shape operators).

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@GB: Just to make sure I understand your question correctly, what you want is a characterization of Riemannian manifolds $M$ such that all the geodesic spheres $S_r=\partial B(p,r)$ around one point $p\in M$ (every point?) have the following property: at every point $x\in S_r$, all the eigenvalues of the second fundamental form of $S_r$ are equal and depend only on $r$. Is that what you are asking for? This would be much much stronger than any warped product -- actually, I suspect it implies $M$ is a space form. –  Renato G Bettiol Dec 10 '12 at 17:00
@GB: Regarding a metric characterization, an equivalent (but silly) way of encoding the above property the doesn't (seem to) use shape operators is to say that "the eigenvalues of the hessian of the distance function $d(x)=dist(p,x)$ are constant on the levelsets of $d$". This is silly because the Hessian of the distance function is precisely the second fundamental form of the geodesic sphere, but might help?... –  Renato G Bettiol Dec 10 '12 at 17:03
@Renato, 1) The condition is hold only for one fixed point say p; 2) Fix the distance sphere $\partial B(p,r)$, then the set of eigenvalues does not depend on the point on this sphere, like the complex projective space case, where the eigenvalues are described in the post above. I was hoping there is some other 'name' for this kind of space. –  J. GE Dec 10 '12 at 20:26
@Renato, note in the complex projective space case, the eigenvalues of a fixed distance sphere is the set $\{\cot(r), \cdots, \cot(r), 2\cot(2r)\}$. i.e. there are $(2n-2)$ $\cot(r)$ in the list. I suspect that this is a kind of weaker 2-point homogeneous space. –  J. GE Dec 10 '12 at 20:32
@GB: Thanks for the explanations. I don't quite see a characterization yet, but for sure all the CROSS (compact rank one symmetric spaces) with their standard Fubini metric satisfy your condition. Namely, $S^n$, $\mathbb CP^n$ and $\mathbb HP^n$, $\mathbb{Ca}P^2$ have a (linear) cohomogeneity one action with a fixed point (the point you call $p$), and the geodesic spheres around $p$ are principal orbits of the action, which eventually collapse to a lower dimensional projective space via the Hopf map. Consequently, these spheres are homogeneous, hence the principal curvatures are constant... –  Renato G Bettiol Dec 10 '12 at 23:10
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