Examples of non-positively Curvature Riemannian Manifolds

Question

When I read about complete, simply connected, and connected Riemannian manifolds of non-positive curvature I only find explicit examples of hyperbolic $n$-space and Euclidean space. What are other commonly used spaces meeting these criteria?

I'm interested in two types of "applications"

1. Used within differential geometry.
   • In this case, I'm most interested in examples of symmetric spaces of non-compact type besides $H^n$ and symmetric positive-definite matrices.
   • Some concrete examples of non-constant and non-positive curvature would be nice.
2. Examples of geometries used in statistics, and applied sciences
   • Gaussian densities I already know about ( which really follows from the former + some information geometric considerations).
   • I leave this up to interpretation, besides saying I'd like to gain some concrete use with my audience. So the hyperboloid example, from the comments, is nice but the examples of symmetric PSD matrices above is more reflective of what I am aiming for.

Edit: I added some comments following Moishe Kohan's remarks and certain other comments.

Thought: I guess we can always generate more "explicit" examples as follows: given any $\phi \in Diff(M,N)$, where $(M,g_M)$ is either of $\mathbb{R}^n,\mathbb{H}^n$, space of symmetric positive definite matrices, and $N$ is some smooth manifold diffeomorphic to $M$, then $g_N:=\phi_{\star}(g_M)$ will give us a non-positively curved Riemannian structure on $N$, since $g_N$ is conformal to $g_M$... Though a direct construction of an example like this is a bit...underwhelming.*

Answer 1

If and when I have more time, I will add further details, the following is just a (pretty long) stub.

First of all, if $(M,g)$ is a complete connected Riemannian manifold of sectional curvature $\le 0$, then lifting $g$ to the universal covering of $M$ results in a Hadamard manifold, i.e. a complete simply connected Riemannian manifold of nonpositive curvature. Note that if $M$ is compact, the metric $g$ is automatically complete. Now, some examples:

1. If $(M_1, g_1), ...., (M_k,g_k)$ are Hadamard manifolds, so is their product $M_1\times ...\times M_k$ equipped with the product-metric $$ g= g_1+...+g_k. $$

From this, you see that products of hyperbolic spaces and Euclidean spaces are Hadamard manifolds.

2. The direct product construction generalizes to certain warped products of manifolds of nonpositive curvature (provided that the warping function is convex), see

Bishop, R. L.; O’Neill, B., Manifolds of negative curvature, Trans. Am. Math. Soc. 145, 1-49 (1969). ZBL0191.52002.

3. As an application of the warping construction M.Anderson proved in

Anderson, Michael T., Metrics of negative curvature on vector bundles, Proc. Am. Math. Soc. 99, 357-363 (1987). ZBL0615.53025.

that if $E\to M$ is a vector bundle over a manifold $M$ admitting a complete metric of nonpositive curvature, then $E$ also admits such a metric.

3. Metrics of nonpositive curvature are frequently constructed by gluing manifolds of nonpositive curvature (with totally-geodesic boundary!) along boundary, provided some boundary conditions are met. Using this, B.Leeb proved in

Leeb, Bernhard, 3-manifolds with(out) metrics of nonpositive curvature, Invent. Math. 122, No. 2, 277-289 (1995). ZBL0840.53031.

that "most" closed aspherical (i.e. with trivial higher homotopy groups) 3-manifolds admit metrics of nonpositive curvature. (Asphericity is clearly a necessary condition.)

4. There are constructions of (complete) metrics on nonpositive curvature on branched coverings over complete Riemannian manifolds of nonpositive curvature (again, provided some conditions on the branch-locus are met), see e.g.

Gromov, Mikhael; Thurston, William P., Pinching constants for hyperbolic manifolds, Invent. Math. 89, 1-12 (1987). ZBL0646.53037.

5. There are "cusp-closing" constructions of metric of nonpositive curvature (starting, say, with complete hyperbolic manifolds of finite volume), see e.g.

Schroeder, Viktor, A cusp closing theorem, Proc. Am. Math. Soc. 106, No. 3, 797-802 (1989). ZBL0678.53034.

6. All symmetric spaces of noncompact type have nonpositive curvature. One can spend whole life studying these, see e.g.

Helgason, Sigurdur, Differential geometry, Lie groups, and symmetric spaces., Graduate Studies in Mathematics. 34. Providence, RI: American Mathematical Society (AMS). xxvi, 641 p. (2001). ZBL0993.53002.

The space $P_n$ of positive-definite symmetric $n\times n$ matrices is just one example of a symmetric space of noncompact type. (Technically speaking, one needs to assume that the determinant equals 1, otherwise, you get the product of ${\mathbb R}$ with a symmetric space of noncompact type.) However, every symmetric space of noncompact type admits an isometric totally-geodesic embeddings in $P_n$ for some $n$, so in this sense, $P_n$ is the main example. A famous algebraic application of nonpositive curvature of symmetric spaces is Cartan's Theorem:

Let $G$ be a connected semisimple Lie group. Then all maximal compact subgroups of $G$ are conjugate to each other.

In my answer I gave (with one exception) no explicit formulae, but all known constructions of metrics of nonpositive curvature are quite explicit. (I am not counting Ricci Flow in dimension 3, since limit metrics have constant curvature.)

As for applications, I still do not entirely understand the question. Having nonpositive curvature has various implications for topology of manifolds, especially their fundamental groups. Would you count decidability of the word problem (for $\pi_1$ of a compact manifold of nonpositive curvature) as an application? I would. As another example: If $M$ is a connected manifold which admits a complete metric of nonpositive curvature, then $\pi_1(M)$ satisfies Novikov's Conjecture. Another famous topological application is in work of Farrell and Jones on Borel conjecture.

Now, if you allow for singular metrics of nonpositive curvature (CAT(0) spaces) then there are applications in mathematical biology and mathematical physics, see e.g. my answer here.

Edit. Here is the sketch of a proof of Cartan's theorem. I think, it is Cartan's original argument. For details, see for instance Helgason's book listed above or

Donaldson, Simon K., Lie algebra theory without algebra, Tschinkel, Yuri (ed.) et al., Algebra, arithmetic, and geometry. In honor of Yu. I. Manin on the occasion of his 70th birthday. Vol. I. Boston, MA: Birkhäuser (ISBN 978-0-8176-4744-5/hbk; 978-0-8176-4745-2/ebook). Progress in Mathematics 269, 549-566 (2009). ZBL1198.22004.

Step 1. Let $G$ be a connected semisimple real Lie group with finite center. Let $K$ be a maximal compact subgroup. Then the quotient $X=G/K$ is simply-connected and has a (left) $G$-invariant complete Riemannian metric of nonpositive curvature (coming from the Killing from on the Lie algebra ${\mathfrak g}$ of $G$ and the associated Cartajn decomposition of ${\mathfrak g}$). By the construction, $G$-stabilizers of points in $X$ are conjugate to $K$.

Step 2. Let $H\subset G$ be a compact subgroup. It acts isometrically on the Hadamard manifold $X$. One then proves:

Theorem. (Cartan's fixed point theorem) If $H$ is a compact group acting continuously and isometrically on a Hadamard manifold $X$, then $H$ fixes a point in $X$.

There are several arguments for proving this, all (that I know) boil down to taking an $H$-orbit $Hx$ in $X$ (which is bounded by compactness of $H$) and defining a "center" of this orbit. This center is either the Chebyshev center (center of the smallest radius ball containing $Hx$) or the barycenter of $Hx$. (This is where one uses nonpositive curvature.)

By the uniqueness and naturality of the center, it will be fixed by $H$.

Applying this to the symmetric space $X=G/K$, one concludes the proof of Cartan's theorem on maximal compact subgroups.