[Math] Origins of Differential Geometry and the Notion of Manifold

Question

The title can potentially lend itself to a very broad discussion, so I'll try to narrow this post down to a few specific questions. I've been studying differential geometry and manifold theory a couple of years now. Over this time the notion of a manifold, as some object that locally looks Euclidean though globally may not be, has become a very comfortable notion for me. However, the definition of a (smooth) manifold is quite abstract, and it's not particularly obvious from the outset why such an object would be of importance. The modern definition as I've come to know it, which can be found in many textbooks (John Lee's book, or this reprint of the definiton), I'm guessing is the byproduct of a long evolution of definitions. It's my guess that many candidate definitions were adopted and discarded until we eventually settled on the one we use today.

1. What is the origin of our modern definition of manifold? As succinctly as possible, what was the evolution of this definition and
   what was the original inspiration for defining such an object?
2. What were the origins of some of the standard manifolds that are used in practice? For instance, I'm certain that some of the objects
   that gave impetus for the definition of a manifold consisted of $\mathbb{S}^2, SO(3)$, and
   $\mathbb{T}^2$, but what are the origins of say the projective spaces
   $\mathbb{RP}^n, \mathbb{CP}^n$, as well as the Grassmann and Stiefel
   manifolds? The role of manifolds in Hamiltonian mechanics?
3. Are there any "vestigial" concepts that were at one time considered important but eventually discarded due to their
   ineffectiveness, or ones that possibly yielded contradictory results?
4. How did the evolution of topology as a subject intermingle with that of differential geometry?
5. What are some good books tracing the history of differential geometry (that is, the evolution of the ideas)? I know of a few math
   history books, including Boyer's
   book,
   but the parts about differential geometry/topology are left almost as
   afterthoughts with the main text dealing with ancient civilizations
   leading up to the calculus.

This is a rather long question, and I don't expect anyone to answer each point in it's entirety. Partial answers are welcome.

Answer 1

[2016-07-25]: Section Differential Geometry added.

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Although OP narrowed down the post, there are still many more important historical facts which should be addressed to adequately answer the question, than I can give in this answer. Nevertheless here are some aspects, which might be interesting.

At least we will see, OP is right when he thinks that many different candidate definitions of manifolds competed to become the most suitable one.

We start with question (5), good books addressing the history of differential geometry/topology.

• A History of Algebraic and Differential Topology 1900 - 1960 by Jean Dieudonné.

  I strongly recommend this book which provides a wealth of historical information as well as technical details. Most of it is regrettably beyond my scope, but it's great for me to get at least a glimpse of the exciting development when going through some parts of the book.

In what follows I focus on OPs question (1) and provide some small samples of text mostly cited verbatim from the book.

As we can read in chapter I, the modern development started with the work of Poincaré. It was his groundbreaking long paper Analysis Situs published in 1895 and followed by five so-called Complements between 1899 and 1905.

• The Work of Poincaré

  [ch 1, § 1.] Concepts and results belonging to algebraic and differential topology may already be noted in the eighteenth and nineteenth centuries, but cannot be said to constitute a mathematical discipline in the usual sense.

  Before Poincaré we should therefore only speak of the prehistory of algebraic topology; ...

But note that topological space has not yet been defined at that time. But some intuitive notion of manifolds was already available.

• Of course, before Frèchet (1906) and Hausdorff (1914) the general notion of topological space had not been defined;

  what had become familiar after the work of Weierstrass and Cantor were the elementary topological notions (open sets, closed sets, neighborhoods, continuous mappings, etc.) in the spaces $\mathbb{R}^n$ and their subspaces;

  these notions had been extended by Riemann (in an intuitive way and without any precise definition) to $n$-dimensional manifolds (or rather what we now would call $C^r$-manifolds with $r\geq 1$).

In this chapter I Dieudonné examines rather detailed Poincaré's Analysis Situs. He explains that Poincaré was the first who introduced the idea of computing with topologial objects, not only with numbers. Most important he introduced the concepts of homology and fundamental group.

With respect to manifolds we also find:

• Poincaré appealed to the concept of oriented manifold by Klein for surfaces and generalized by von Dyck for manifolds of arbitrary dimension. In Analysis Situs, Poincaré gave a characterization of orientable manifolds by what is still one of the modern criteria:

  there exist charts $(U_\lambda,\psi_{\lambda})$ such that the transitition diffeomorphisms $$\psi_{\lambda}\left(U_\lambda\cap U_{\mu}\right)\rightarrow\psi_{\mu}\left(U_{\lambda}\cap U_{\mu}\right)$$ have positive determinants for all pairs of indices such that $U_{\lambda}\cap U_{\mu}\neq \emptyset$.

But later on Dieudonné addresses also some weak points in connection with this definition. In fact we won't find some final definition of the term manifold by Poincaré as we can read in the next paragraph. Nevertheless the far-reaching character of his work is tremendous:

• ... As in so many of his papers, he gave free rein to his imaginative powers and his extraordinary intuition, which only very seldom led him astray; in almost every section is an original idea.

  But we should not look for precise definitions, and it is often necessary to guess what he had in mind by interpreting the context. ...

  Thus ends this fascinating and exasperating paper, which, in spite of its shortcomings, contains the germs of most of the developments of homology during the next 30 years.

In section I. § 4 Duality and Intersection Theory on Manifolds there is a subsection $A$ titled with

The Notion of Manifold

• [ch 1, § 4.A] After the invariance problem had been solved, two main items remained in the implementation of the program outlined by Poincaré: a rigorous proof of the duality theorem and a complete theory of intersection barely begun by Poincaré.

  Obvious examples show that in neither question can one work with a general cell complex; some restrictions have to be introduced in order to make available the arguments Poincaré used for his manifolds.

  ...

  In the meantime, in order to use simplicial methods, topologists had to settle for more tractable definitions of manifolds. In fact, several definitions were proposed ([308], pp. 342-343);

Dieudonné refers here to Algebraic Topology by S. Lefschetz from 1942. We can find there $9$ different types of manifolds, all of which are supposed to be $n$-dimensional.

• Combinatorial manifolds: Let $X=Y-Z$ be an open simple complex, where $Y$ is closed simple and $Z$ is a closed subcomplex of $Y$. We say that $X$ is an orientable combinatorial manifold whenever the following two conditions are fulfilled.

  (1) The dual $X^\star$ of $X$ has a closed simple weak isomorph $\overline{X}$.

  (2) If $x,x^\prime$ are distinct elements of $X$, then $\text{Cl } x \cap \text{St } x^\prime$ is acyclic or void.

  They may be finite or infinite, absolute or relative, orientable or non-orientable, simplicial or merely simple complexes.

• Geometric manifolds: Euclidean realizations of the preceding simplicial types.

• Manifolds in the sense of Brouwer: Euclidean complexes such that the star of each vertex is isomorphic with a set of simplexes in an Euclidean $\mathcal{E}^n$ having a common vertex $P$ and making up a neighborhood of $P\in\mathcal{E}^n$.

• Manifolds in the sense of Newman: Euclidean complexes such that if $a$ is a vertex $\text{St} a = aB$, then $B$ is partition-equivalent to an $(n-1)$-sphere.

• Manifolds in the sense of Poincaré and Veblen: Topological complexes such that every point has for neighborhood an $n$-cell.

• Topological manifolds: An $M^n$ of this type is separable metric space with a countable locally finite open covering consisting of $n$-cells. Noteworthy special cases: $C^r$-manifolds, differentiable manifolds, analytical manifolds, $\Gamma$-manifolds, group manifolds.

• Generalized manifolds: Locally compact spaces discussed by Čech, Lefschetz, Wilder and others and characterized by certain properties of so-called local connectedness or local connectedness in the sense of homology and also by the property: each point is $n$-cyclic.

• Pseudo-manifolds: This term has been applied by Brouwer and other authors to what we have called a simple geometric $n$-circuit.

• Manifolds of grade $p$: Simplicial $n$-complexes, investigated by Čech, and which behave like an $M^n$ only as regards the two consecutive dimensions $p-1,p$.

Of course this answer hardly touches the surface of information provided in J. Dieudonné's book. ... curious? :-)

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A nice historical survey is The Concept of Manifold, 1850-1950 by Erhard Scholz. Section 5 is devoted to the development of the modern manifold concept. In subsection 5.4 he describes the birth of the "modern" axiomatic concept in differential geometry.

Differential Geometry

There was, of course still another line of research, more closely linked to differential geometry, where manifolds played an essential role, and purely topological aspects (independently of whether continuous, combinatorial, or homological ones) did not suffice and still needed elaboration. ...

and later on

Whitehead and Veblen presented their axiomatic characterization of manifold of class $G$ first in a research article in the Annals of Mathematics (1931) and in the final form in their tract on the Foundations of Differential Geometry (1932).

Their book contributed effectively to a conceptual standardization of modern differential geometry, including not only the basic concepts of continuous and differentiable manifold of different classes, but also the modern reconstruction of the differentials $dx=(dx_1,\ldots,dx_n)$ as objects on tangent spaces to $M$.

Basic concepts like Riemannian metric, affine connection, holonomy group, covering manifolds, etc. followed in a formal and symbolic precision that even from the strict logical standards of the 1930-s there remained no doubt about the wellfoundedness of differential geometry in manifolds.