On a definition of manifold

Solution 1:

You've stumbled across one of the interesting pitfalls of the history of science.

Those words attributed to Riemann should not be understood as equivalent to the modern definition that you have learned for a manifold, i.e. as "something glued by locally homemorphic to Euclidean parts".

Riemann's development of the concept of a manifold was far beyond that of any predecessors. The words you quoted are, most likely, some translation of words that Riemann concocted to try to explain this concept to his contemporaries in an intuitive manner that made sense to them.

If 150 years of hindsight are ignored, it might be easy to forget that as far as Riemann got in understanding the concept of manifolds, he did not get as far as the concept that we use in our modern differential topology classes. In fact, I believe that the modern language of atlases --- coordinate charts and smooth overlap maps --- was not used in its full, abstract form until well into the 20th century, perhaps not until John Milnor's early lectures on differential topology.

Solution 2:

Ok. First of all, I'm not entirely sure what Riemann meant by that. I am going to give you a standard definition of smooth manifolds using the language that is most commonly used in mathematics today. I also include some more thoughts after the definitions.

Topological Manifolds

Before talking about what a smooth manifold is (one on which you can perform calculus), we should be clear about what properties we would like our space to have. For this we have that a topological manifold $M$ is a topological space consisting of the following properties:

Locally Euclidean - for every point $p \in M$ there is a neighborhood $U \subseteq M$ containing $p$ and a homeomorphism $f: U \to V$ where $V \subseteq \mathbb{R}^n$ is an open subset. It is often convenient to restrict to a connected subset $U$, then we can take $V = \mathbb{R}^n$ itself.
Hausdorff Property - distinct points of $M$ can be separated by disjoint open sets (this property is important in making sure that sequence of points in $M$ converge to unique limits).
Existence of Countable Topological Basis - There exists a countable open cover $\mathcal{B} = \{B_i \subseteq M \; | \; i \in I \}$ such that whenever $B_i \cap B_j \neq \emptyset$ there exists some $B_k \in \mathcal{B}$ such that $B_k \subseteq B_i \cap B_j$ (this property is needed to define a structure called a partition of unity which is often used to "glue" locally-defined functions, forms, and vector fields into globally defined objects).

Smooth Manifolds

In order for one to start doing geometry on manifolds we need something called a "smooth structure", which takes some care to develop. Without loss of generality, we will assume that when we talk about homeomorphic neighborhoods we'll simply use $\mathbb{R}^n$ instead of some arbitrary open Euclidean subset. We also assume $M$ is a topological manifold. We'll take this in steps:

A pairing of an open subset $U \subseteq M$ and a homeomorphism $\varphi:U \to \mathbb{R}^n$ is called a chart for $M$ and we write this as $(U, \varphi)$.
An atlas for $M$ is a collection of charts $\mathcal{A} = \{(U_\alpha, \varphi_\alpha)\}$ such that $\bigcup_\alpha U_\alpha = M$.
An atlas $\mathcal{A}$ for $M$ is said to be a smooth atlas whenever $(U_\alpha, \varphi_\alpha)$ and $(U_\beta, \varphi_\beta)$ are charts in $\mathcal{A}$ we have that $\varphi_\beta \circ \varphi_\alpha^{-1}$ is a smooth map between Euclidean spaces provided $U_\alpha \cap U_\beta \neq \emptyset$ (in fact, they are actually diffeomorphisms).
A differentiable structure for the manifold $M$ is a maximal smooth atlas (meaning a smooth atlas which is not contained in any other smooth atlas; this is mostly used as a theoretical tool for covering all possible atlases that are smoothly compatible with one another). A pairing of a manifold $M$ with a differentiable structure is called a differentiable manifold or smooth manifold.

While none of this necessarily explains Riemann's terminology, this gives one way authors typically define what a smooth manifold is. Manifolds are often used to describe state spaces, or spaces of information for a system (oftentimes in physics, data analysis, and now some computer programming). They are defined in such a way that they become the most general spaces on which one can perform calculus. Spaces which are usually of interest in the world, on which one would like to study smooth variations of a quantity, more often than not exhibit a non-Euclidean structure. Typically too, while a manifold usually represents the space of all possible states for a system, the actual states a system occupies usually traces out a curve on some manifold. "Transition of an instance" may possibly have to do with how the variation of the state of a system can be mapped as some time-dependent function on a non-Euclidean space.

Solution 3:

What's written is pretty much meaningless. If you are interested in a modern interpretation of Riemann's lecture, my suggestion is to read Spivak's "A Comprehensive Introduction to Differential Geometry," Vol. 2, where he discusses the mathematical meaning of Riemann's lecture in great detail (pages 163-180, What did Riemann say?).

Solution 4:

I think your authors misquoted Riemann. In his famous 1854 essay, Riemann mentions the possibility of passing continuously from one "mode of determination" to another, by which he means, in modern terminology, that the quadratic form determining the metric varies continuously from point to point in a manifold.

Thus he did not say "continuous transition of an instance" but rather "continuous trandition of a mode of determination" the latter being a quadratic form in modern terminology.

Solution 5:

Well, I'm not exactly sure what Riemann had in mind, but if I had to guess it has to do with the principal of locality. Since Riemann appears to be explaining by way of analogy, by appealing to a physical (or perhaps metaphysical) concept like an instance (or maybe a moment), I'll continue with the analogy by appealing to physical concepts.

Almost all of physics can be broken into two different studies: that of dynamics and that of kinematics. Dynamics is the study of objects under the influence of causes, such as a particle moving in the presence of an electromagnetic field, or the motion of a ball after it has been pushed. On the other hand kinematics seeks to study the motion of objects without reference to causes.

So what could be meant by instance, is really the moment at which something affects another thing. The Euclidean model is very good at describing the dynamics of an event. Generally, one can describe the dynamics by imposing a coordinate system and describing the position of using the coordinates. One can then describe the motion of the object in these coordinates after some cause affects the object.

On the other hand, in the presence of nontrivial kinematics, the full effect of the cause is more difficult to describe. The idea is that motion of the object itself is subject to some nontrivial geometry independent of the cause.

Think of kinematics as being described by manifolds: the bare geometry on which points live; think of dynamics as modelling motion under the influence of causes without reference to this geometry. In this analogy, what manifolds do is allow us to describe both the kinematics and dynamics of an event locally. A local coordinate chart tells us how the dynamics at each point changes, as we vary the point in a continuous fashion...so sort of a continuous transition of an instance.

Of course, this is all guesswork. Since I have no idea what Riemann really meant by that statement.