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"The answer is that this Spherical Geometry was merely considered to be inherited from the Euclidean geometry of the 3-dimensional space in which the sphere resides. No thought was given in those times to the sphere's internal 2- dimensional geometry as representing an alternative to the Euclid's plane. Not only did it violate Euclid's fifth axiom, it also violated a much more basic one (Euclid's first axiom) that we can always draw a unique straight line connecting two points , for this fails when two points are antipodal.

Tristan Needham VDG Pg-7, sect.1.2 Spherical Geometry

What does it mean for Euclidean geometry of the sphere to be inherited from 3-d Space? I can't quite understand the language of the above paragraph.

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Clemens Bartholdy
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    It dies not say that Euclidean geometry is inherited from 3-d Space. It says that consideration of the spheres geometry only thought of the sphere as being an object within 3D space. The next 2 sentences tell you what was not considered - the inherent geometry of the sphere surface itself, independent of its embedding in 3D space. – Paul Nov 26 '21 at 08:26
  • I don't get it, what really is the difference between there being an ambient space or not? @Paul – Clemens Bartholdy Nov 26 '21 at 08:27
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    Difference in what sense? According to general relativity, our own ambient space is not Euclidean, we only think it is locally. If we were sphere livers with no sense of 'up' we would have developed a 2D local Euclidean Geometry. Later polar expeditions might make us then re-consider our notion of what geometry is. We would need an intrinsic geometry of our universe, the sphere. – Paul Nov 26 '21 at 08:42
  • As pointed out, the question is wrong - nothing like this is implied by the paragraph. Instead, maybe you meant to ask what does it mean for spherical geometry to be inherited from the Euclidean 3-dimensional space? Please consider correcting your question. – Michał Miśkiewicz Nov 26 '21 at 15:55

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In this setting, when we speak of "geometry", we're talking about a choice of a Riemannian metric on a smooth manifold. I'm sure you can look up the precise definitions yourself, but the idea is that Riemannian metrics are what define the notion of a geometry on a given space; they prescribe at "an infinitesimal level" (i.e at the level of tangent spaces) an inner product, meaning a notion of lengths and angles (once we have an "infinitesimal" description of lengths and angles, we can build up to a general notion of distances, areas, volumes etc, i.e all the basic "geometric" notions).

To say spherical geometry is "inherited" from the Euclidean geometry in $\Bbb{R}^3$ means the following. Let $(x,y,z)$ be the Cartesian coordinates in $\Bbb{R}^3$ (i.e the identity chart) and define $g=dx\otimes dx+dy\otimes dy+dz\otimes dz$, then $g$ is a Riemannian metric on $\Bbb{R}^3$. Now, let $\iota:S^2\to \Bbb{R}^3$ be the canonical inclusion mapping. Then, we can consider the pullback tensor field $\iota^*g$ on $S^2$. Since $\iota$ is an immersion (i.e at every point $p\in S^2$ its tangent linear mapping $T\iota_p:T_pS^2\to T_{\iota(p)}\Bbb{R}^3=T_p\Bbb{R}^3$ is injective) it follows that $\iota^*g$ is indeed a Riemannian metric on $S^2$. This is what we refer to as the Riemannian metric "induced" on $S^2$.

More generally, if you start with a Riemannian manifold $(M,g)$ and you consider an embedded submanifold $S\subset M$, then we can look at the inclusion $\iota:S\to M$, and then look at the pullback $\iota^*g$. This way $(S,\iota^*g)$ is a Riemannian manifold in its own right (i.e it has its own notion of lengths and angles "on an infinitesimal level"), and we call this a Riemannian submanifold of $(M,g)$. We also say that $S$ inherits its geometry from $M$ (more precisely from $(M,g)$).

I'm sure this terminology of tensor fields, smooth manifolds, pullbacks etc has not been introduced by page 7 of the book, but that's what is meant (even if at this stage of the book one does not have access to this terminology). Also, one of course has to define what "straight lines" mean in this general context, but you'll get to all of this as you continue reading.

peek-a-boo
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