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I will start with some definitions and then present the first half of chapter II.2. in ``Über die Hypothesen, welche der geometrie zugrundeliegen".
Riemann's defines the notion of a manifold inductively as follows:
In a concept whose instances form a continuous manifold, if one passes from one instance to another in a well-determined way, the instances through which one has passed form a simply extended manifold, whose essential characteristic is, that from any point in it a continuous movement is possible in only two directions, forwards and backwards. If one imagines that this manifold passes to another, completely different one, and once again in a well-determined way, that is, so that every point passes to a well-determined point of tha other, then the instances fom. similarly a double extended manifold. In a similar way, one obtains a triply extended manifold when one imagines that a doubly extended one passes in a well-determined way to a completely different one, and it is easy to see how one can continue this construction. (...) [Riemann I.2]
The modern definition says that a manifold has to be locally isomorphic to and it is quite likely that Riemann already was aware of this property of manifolds. [Spivak, 4B-2]
On the manifold, Riemann introduces a metric:
...ds equals the squareroot of an everywhere positive homogenuous function of the second degree in the quantities dx in which the copefficients are continuous functions of the quantities x.
So he introduces a norm by , where . The inner product from which is comes is called the Riemannian metric on a manifold M and a manifold with a Riemannian metric is a Riemannian Manifold. From now on every manifold will be Riemannian.
For this purpose, one constructs the system of shortest lines emanating from a given point; the position of an arbitrary point can be determined by the initial direction of the shortest line in which it lies, and its distance, in this line, from the initial point. It can therefore be expressed by the ratios of the quantities , i.e., the quantities dx at the origin of this shortest line and by the length s of this line.
Riemann fixes a point p on his manifold and looks for the location of another point. He takes the the tangent vector of the Geodesic, the shortest line on M between p and q and then the length of this line between p and q. Therefore he needs an vector, and the length.
In place of the one now introduces linear expressions formed from them in such a way that the initial value of the square of the line element will be equal to the sum of the squares of these expressions, so that the independent variables are: the quantity s and the ratios of the quantities . Finally, in place of the choose quantities proportional to them, but such that the sum of their squares equals .
He there diagonalizes the matrix with and gets a diagonal matrix . Finally he takes , such that and denotes by .
If one introduces these quantities, then for infinitely small values of x the square of the line element equals , but the next order term in its expansion equals a homogenous expression of the second degree in the quantities and is consequently an infinitely small quantity of the fourth order, so that one obtains a finite quantity if one divides it by the square of the infinitely small triangle at whose vertices the variables have the values . This quantity remains the same as long as the quantities x and dx are contained in the same binary linear forms, or as long as the two shortest lines from the initial point to x and from the initial point to dx remain in the same surface element, and therefore depends only on the position and direction of that element. It obviously becomes zero if the manifold in question is flat, i.e., if the square of the line element is reducible to , and can therefore be regarded as the measure of deviation from flatness in this surface direction at this point. When multiplies by it becomes equal to the quantity which Privy Councillor Gauss has called the curvature of a surface. [Riemann, 4A-11/12]
The main part is dealing with the second part of the Taylor expansion of the functions he uses to define his norm. The corresponding part in the norm he says to be equal to . Then he defines a quantity by dividing this sum by the square of the triangle spanned by and . He states, that this quantity is not influenced by a linear transformation of the vectors. Because if , there is no second term in the Taylor expansion, this quantity is zero in this case. This is the only ``Proof" he gives that this quantity has to be equal to times the Gaussian curvature, which was defined by Gauss as , where R is a subset of M and .[Spivak, p. 3A-4]
Now I will outline the proof of this theorem in modern mathematics. (I hope sometime, at the moment i will only outline the modern mathematics needed to state the statement.) Therefore we need the Riemannian normal coordinates which he introduces in the first part. One chooses an othonormal basis in the tangent space of the fixed point p. Then a coordinate system is defined by
Then one takes the vector on equal to the tangent vector at p of the geodesic between p and q, with the length of the distance between p and q through the geodesic and by this gets a coordinate system of a neighborhood of p. It is not uniquly determined, but depends of the choice of the ONB. If we compute the Taylor expansion of the composition of the functions of the Riemannian metric with the inversion of the coordinate system , we have
and since , we get
Spivak then states some Propositions about these terms.
1. In a Riemannian normal coordinate system x at p we have
3. Let Q be a quadratic function of 2n variables,
for all matrices A if and only if:
4. A quadratic function
with (1) of 3. satisfies the two equivalent consitions of 3. if and only if it can be written as
5. In a Riemannian normal coordinate system x at p, the numbers
Therefore by 4. it can be written in the way Riemann asserts and one can define a number Q(W) for any 2-dimensional subspace by
with any basis for W. [p.4B-24]
The theorem can then be stated like this: If M is 2-dimensional and , then
(Since Spivak divides by the square of the parallelogram instead of the area of the triangle spanned by the vectors like Riemann, he does not need the factor .)