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Integral Cohomology and Geometry

Geometric Realization of Second Integral Cohomology Classes The second integral cohomology classes in \( H^{2}(M ; Z) \) are in natural \( 1-1 \) correspondance with the isomorphism classes of circle-bundles over \( M \), where $M$ is a manifold. In particular, every second integral cohomology class \( \hat{F} \in H^{2}(M ; Z) \) has a geometric realization which consists of a space \( P=P_{F} \) together with an action of the circle \( S^{1} \) on \(P \) and a natural projection: \[\begin{array} \\ P \\ \downarrow \pi \\ M \end{array}\] so that the \( S^{1} \) action on \( P \) is free, preserves \( \pi, \) and \( \pi \) induces an isomorphism of \( P / S^{1} \) with \( M .\) For instance if \( M \) is a Riemann surface, consider that $P$ is the set of its unit tangent vectors to $M$, and let \( \pi \) be the projection map which assigns to each tangent vector the corresponding point of tangency. The circle \( S^{1} \) then acts freely by rotating any vector counter-clockwise in its...
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Projectivity and Spectrality

Spectral Decomposition with Respect to a Circle Consider that \( T \) is an endomorphism of a finite-dimensional vector space \( E \), and let \( S \) be a circle in the complex plane which does not pass through any eigenvalue of \( T \). Then we consider the operator: \[\mathrm{Q}=\frac{1}{2 \pi i} \int_{S}(z-T)^{-1} d z\] $\mathrm Q$ is a projection operator in \( E \) which commutes with \( T . \) The decomposition \[E=E_{+} \oplus E_{-}, \quad E_{+}=Q E, \quad E_{-}=(1-Q) E\] is therefore invariant under \( T \), so that we can write: \[T=T_{+} \oplus T_{-}\] Then \( T_{+} \) has all eigenvalues inside \( S \) while \( T_{-} \) has all eigenvalues outside \( S \). This is just the spectral decomposition of \( T \) corresponding to the two components of the complement of the circle \( S \).
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Embracing Analytically and Winding

Complex Logarithm and Winding For any path \( \mathcal{C} \)  and any point \( a \) not on \( \mathrm {C}, \) the complex logarithm enables us to define the following function: \[\begin{array}{l} \quad N(\mathcal{C} ; a)=\frac{1}{2 \pi i} \int_{\mathrm{C}} \frac{d z}{z-a}=\frac{1}{2 \pi i} \int_{z} d \log (z-a)=\left.\frac{1}{2 \pi} \arg (z-a)\right|_{\mathrm C} \end{array}\] This is called the winding number of the path \( \mathrm{C} \) with respect to \( a \).  Since the indeterminacy of the argument is in multiples of \( 2 \pi \), if \( \mathrm{C} \) is a closed path, then the values of \( N(\mathrm{C} ; a) \) are integers. The expression \( N(\mathrm{C} ; a) \) is simultaneously a function of both \( \mathrm{C} \) and \( a \). Evidently, \( N \) depends continuously on \( a \) and $\mathrm C$ unless \( a \) reaches \( \mathrm{C} . \) This means that if a family of paths \( \mathcal{C}_{t} \), where none of them is passing through \( a \), were parametrized continuously b...
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Analyticity and Integrality

Analyticity and the Complex Logarithm We consider a complex-valued function $f$ of one complex variable $z$, and we write: \[ \int_{\mathrm{A}} f(z) d z=(\mathrm{A}, f(z) d z)\] interpreting it as being bilinear in the differential \( f(z) d z \) and the homological $1$-chain \( \mathrm {A} \). Then, if \( f \) is a continuous function of the complex variable \( z \) in the region $\mathbf R$ and if it holds: \[\int_{\mathrm{C}} f(z) d z=0\] about any closed chain $\mathrm {C}$ in \( \mathbf{R}, \) then \( f(z) \) represents an analytic function in $\mathbf {R}$. In practice, if \( f \) is analytic merely on a closed path \( \mathcal{C}, \) it does not follow that \( \int_{\mathrm{C}} f(z) d z=0 . \) In particular, \[\int_{\mathrm{C}} \frac{d z}{z}=2 \pi i\] where \( \mathcal{C} \) is a simple closed curve embracing the origin. This leads to the conclusion that it is precisely the complex logarithm function that characterizes the geometry of the complex plane in all its essential aspe...
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Archimedean Metronomy II

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Archimedean Area Metronomy by Descent The area of a  spherical dome is equal to the area of a disk, the radius of which is the length of the segment joining the top vertex of the dome to its boundary. Equivalently, if $S$ is this vertex of the spherical dome and $A$ any point on the boundary circle, then the area of the spherical dome is equal to the area of a disk with radius $SA$. If the spherical dome is the whole sphere, then $S$ and $A$ are antipodes, and thus the area of the sphere is $\pi (SA)^2=4 \pi \rho^2$, where $\rho=SA/2$. \[S C=h, O C=R-h, O A=R\] Hence, we have: \[A C^{2}=R^{2}-(R-h)^{2}=2 R h-h^{2}\] \[S A^{2}=A C^{2}+C S^{2}=2 R h\] Thus, we identify $SA$ with the geometric mean of $2SO$ and $SC$: \[S A=\sqrt{2 S O \cdot S C}\] We conclude that the area of a spherical dome is measured by descending to the area of the corresponding disk in a measure-preserving manner. Let \( A_{1}, A_{2} \) be two points on the half-sphere; we consider the domes delimited by the p...
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Imaginary Transformation à la Klein

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Imaginary Circular Points and the Rectangular Hyperbola Consider the equation of the circle: \[(x-a)^{2}+(y-b)^{2}=r^{2}\] which can be written in the homogeneous form: \[(\xi-a \tau)^{2}+(\eta-b \tau)^{2}-r^{2} \tau^{2}=0\] The intersection with the line at infinity \( \tau=0 \) will thus be given by the equations: \[\xi^{2}+\eta^{2}=0, \quad \tau=0\] The constants \( a, b, \) and \( r, \) which characterise the present circle, do not appear at all. Hence, every circle cuts the line at infinity in the same two fixed points: \[\xi: \eta=\pm i, \quad \tau=0\] which are called the imaginary circular points. The converse is also true: Every curve of second degree, which passes through the imaginary circular points in its plane is a circle. The distance from the origin to the imaginary circular points is not definitely infinite, as might perhaps at first be thought of. Instead, this distance has the form:  \[ \sqrt{x^{2}+y^{2}}=\sqrt{\xi^{2}+\eta^{2}} / \tau=0 / 0 \]  and is con...
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Archimedean Metronomy

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Spiral Bridge of Rectification Archimedes thinks of the spiral in terms of the combined mechanical motion of a point undergoing two distinct uniform motions: A uniform motion in a fixed linear direction with constant velocity, and a uniform motion in a circle with constant velocity as well. Both uniform motions start at the same point, which is considered as the pole of the non-uniform combined motion traced by the emerging spiral. The spiral is depicted together with its tangent attached to each point of the motion it traces. After a complete turn, the area of the circle is rectified, through the spiral bridge, in terms of the area of the adjacent to the spiral triangle emerging by means of the attached tangent, and thus the quadrature of the circle follows. The perimeter of the circle is metronomically unrolled to the side of the orthogonal triangle opposite to the angle of the tangent, whose other side is the radius of the circle.
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