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Is this true that every simple closed curve on the earth can be deformed continuously to a point without leaving the earth?

Is the earth compact?

Now if we consider the earth as a 2-manifold, can we say that the earth is a sphere by the classification of 2-manifolds?

What is the reason? Is there a practical way to detect the above properties for the earth?

Edit: With the assumption that the earth is simply connected (without hole), the earth must be a sphere or a plane (up to the compactedness of the earth).

  • Of course the earth is not a manifold. A manifold is a mathematical object and the earth is not. And the earth, not being a mathematical object,is neither simply connected nor compact. You can model the earth as a mathematical object, and then the question of whether it's simply connected, compact, etc, depends on the model you've chosen. Nobody can answer this question without knowing what model you've chosen. Once you choose a model, this will probably be a very easy question to answer --- but only if you reveal your choice. – WillO Apr 28 '19 at 06:10

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We typically take the Earth to be a 2-manifold.

If we want to be as truthful as possible, it's best modeled as a 3 dimensional shape, upon which we can define a metric which lets us define a 2-manifold. For all intents and purposes, that assumption is pretty good.

However, sometimes the most straightforward answer is a counterxample:

Moab

Cort Ammon
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Near the village where I grew up, Wookey Hole, there is a cave system called (rather unimaginatively) the Wookey Hole Cave. The areas mapped so far look like this:

Wookey Hole Cave

The part of the cave between sump 12 on the left and chamber 25 on the right has not been mapped but we know the two side are connected because dye poured into Swildon's Hole emerges at the resurgence.

Anyhow the existence of this cave system means the Earth is not topologically equivalent to a 2 sphere since a circle drawn around either cave entrance cannot be contracted to a point, not that I have tried this since the authorities take a dim view of people painting circles around the cave entrance.

John Rennie
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  • There are many caves in the Earth, so the surface of the Earth is multiply connected. I have no idea what the genus of the Earth's surface is, but it will be a very high number. – John Rennie Apr 28 '19 at 06:10
  • @ john So you say that the earth is similar to the torus (or g-torus more generally). –  Apr 28 '19 at 05:09
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    The woodchuck burrow under my porch has multiple entrances, and might be a more accessible example for anyone who wants to see this phenomenon with his own eyes.. – WillO Apr 28 '19 at 05:04
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If you're considering the Earth as a 2-manifold from the beginning, yeah, one can deform continuously any closed simple curve into a point. If you are really taking something as similar as possible to earth surface with it's imperfections, then no, we can't, because it's not smooth.

The Earth would be topologically equivalent to a sphere, by being a compact smooth manifold with no boundary and no holes. But it's more like a revolution ellipsoid (Ref: https://en.wikipedia.org/wiki/Figure_of_the_Earth). It has non-zero quadrupole gravitational moment, which would be zero for a uniform sphere.

Gabriel Picanço
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  • By the first part, I mean whether the topology of the earth is actualy simply connected. That is, can we prove this experimentally? –  Apr 28 '19 at 03:52
  • Maybe the earth is similar to torus and has a hole. –  Apr 28 '19 at 03:55