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I have been reading introductory material on differential geometry and have come across the term flatness a lot.

Intuitively a layman would assume "flatness" means that the underlying manifold is flat-shaped (like a hyper-plane), but from my readings I don't believe that is the case. I believe you can have many shapes of manifolds which are also "flat", according to the differential geometric definitions.

I believe "flatness" refers solely to having a covariant derivative (a connection) which has zero torsion (Christoffel symbols evaluate to zero).

Thus to help solidify my intuition I'm hoping someone can help 1. Verify what I have written thus far, and 2. Answer the following questions:

  1. Does flatness have any implication for the shape of the underlying manifold?

  2. If no, then does the property of flatness exist on any manifold we choose? Since flatness seems to depend on having an "affine connection" which can be constructed without explicit reference to any intrinsic shape of the manifold (and by extension there exist infinite types of parallel transport - so what's stopping me from always selecting zero curvature covariant derivatives)

  3. Is flatness a desirable attribute and or property of an underlying manifold?

  4. In Amari's work on information geometry, he defines a notion of dual flatness on a statistical manifold. Would dual flatness change any of the interpretations of the previous answers?

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  • $\begingroup$ One example to always keep in mind is the flat torus $(\Pi^2,g_{\text{flat}})$ - it has the exact same shape as the embedded torus (i.e. they are the same as smooth manifolds), but is equipped with a "pushforward" metric coming from the universal cover $\mathbb{R}^2 \to \Pi^2$, where $\mathbb{R}^2$ is equipped with the standard Euclidean Riemannian metric. In this case the sectional curvature of $(\Pi^2,g_{\text{flat}})$ is $0$, thus giving an example where the Riemannian metric forces flatness and contradicts your intuition. $\endgroup$
    – noctusraid
    Sep 24 '19 at 10:48
  • $\begingroup$ But can we not also asign a different connection over the manifold, which can (I think) lead to a different metric? In this way can the defn of curvature be arbitrary, since it is dependent on the appearance of the connection/covariant derivative. Therefore is curvature something we can choose, does it exist independently of what manifold we are working with? $\endgroup$ Sep 24 '19 at 12:03
  • $\begingroup$ My intuition on arbitrariness comes from the fact that there are infinite connections and parallel transports we can define over a manifold (I think), so it seems we can just construct it howverte we want it. $\endgroup$ Sep 24 '19 at 12:04
  • $\begingroup$ There are many different geometries (metrics/connections) that can be put on the same topology/manifold (I think this corresponds to your "shape"). There are many connections between geometry and topology; for example, the Gauss-Bonnet theorem tells us that some surfaces cannot be flat. The study of these connections makes up a large part of differential geometry. $\endgroup$ Sep 24 '19 at 12:45
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    $\begingroup$ The term "flat" refers to the curvature of the connection — it means zero curvature. The Gauss-Bonnet Theorem, as @AnthonyCarapetis mentioned, relates the curvature of a compact, even-dimensional Riemannian manifold (with the canonical Levi-Civita connection) to its topology. There are various generalizations of this theorem to more general vector bundles/connections. $\endgroup$ Sep 25 '19 at 0:01
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A connection is a procedure for transporting geometric data across curves. An affine connection is a procedure for transporting tangent vectors specifically. More concretely, given any smooth curve $\gamma:[0,1]\to M$ and any vector $v\in T_{\gamma(0)}M$, an affine connection defines a unique vector $v'\in T_{\gamma(1)}$, which we can treat as in some sense "the same as" (or "parallel to") $v$. For this reason we call $v'$ the parallel transportation of $v$ along $\gamma$. We can also define a connection an affine connection in terms of a covariant derivative: the parallel transportation of $v$ is then the unique solution of $\nabla_{\dot\gamma(t)}v(t)=0$.

"Flatness" as it is used in differential geometry, is a property of connections. A connection is flat if parallel transportation along any closed curve is the identity. Equivalently, a connection is flat if parallel transportation is path independent. We can also define local flatness by applying the same criteria to paths contained in a neighborhood of each point. When we say a Riemannian manifold is flat, we mean that the Levy-Civita connection (which is uniqely determined by the mertic) is flat.

$\mathbb{R}^n$, as a Riemannian manifold with the standard metric, is flat. It is for this reason that we often don't think of tangent vectors in Euclidean space as as attached to points: we can freely identify them with tangent vectors at the origin without having to choose a specific path.

We say a Riemannian manifold $(M,g)$ is locally flat at $p$ if any of the following apply:

  • The Levi-Civita connection on $M$ is locally flat at $p$.
  • There is a neighborhood of $p$ which is isometric to a neighborhood in $\mathbb{R}^n$
  • There is a set of local coordinates around $p$ in which the metric is just the. euclidean metric $g_{ij}=\delta_{ij}$ on the entire coordinate patch.
  • There is a set of local coordinates in which the Christoffel symbols are identically zero on the entire coordinate patch.
  • The Riemann curvature tensor is identically zero on a neighborhood of $p$.

On a manifold, $M$ there are many choices of metric, each of which comes with a Levi-Civita connection. The existence of a flat metric comes with several topological restrictions, and there are many manifolds which admit no flat metrics.

As an example, the 2-Sphere $S_2$ has no flat metrics: On a flat Riemannian manifold, we can extend every nonzero tangent vector to a smooth, nonvanishing vector field by transporting the tangent vector to every other point (the fact we can do this smoothly depends on path independence). On $S_2$, there are no such vector fields (by the hairy ball theorem). Therefore, $S_2$ does not admit a flat Riemannian structure. Additionally, since the $S_2$ is simply connected, all locally flat connections are globally flat, so it doesn't even admit a locally flat Riemannian structure.

There are also manifolds (such as the Möbius strip) which can be equipped with a locally flat Riemannian structure, but not a globally flat one, again due to topological obstrucions.

If we instead define general affine connections, we have more freedom to choose connection with torsion (the LC connection is by definition torsion free). In this case, there are two obstructions to local flatness: the Riemann tensor and the torsion tensor must be identically zero. This formulation of your question isn't that different, though, since every flat affine connection is a LC connection for some metric.

To address your question, then, the fact that a Riemannian manifold is (locally) flat can tell you quite a bit about its topology, but it doesn't uniquely determine it. Conversely, we can only construct flat affine connections on manifolds with specific topologies.

I'm not familiar enough with statistical manifolds to comment on your last point.

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    $\begingroup$ Hey thanks for the lovely answer. $\endgroup$ Sep 26 '19 at 11:18

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