Topological surfaces and Riemannian manifolds
Introduction
The topological approach to the study of surfaces reveals properties that are invariant with respect to continuous deformations, taking geometry beyond rigid metric relations into the field of qualitative structural characteristics. In this example, classical non-orientable surfaces are considered, as well as multi-leaf structures of complex functions are visualized. The software implementation using interactive three-dimensional constructions makes it possible to visualize the difference between local behavior in the vicinity of regular points and global topological features such as self-intersections and loss of orientation.
The Mobius Strip
The Mobius strip is the simplest non-orientable surface obtained by gluing the opposite sides of a rectangle with a preliminary half-turn. A surface has a single side and a single boundary curve, which makes it a fundamental counterexample in the topology of manifolds. When traversing along the midline, the normal vector reverses direction, demonstrating a loss of global orientability while maintaining the local Euclidean structure.
Parametric equations of the classical implementation:
where .
u = range(0, 2π, length=100)
v = range(-1, 1, length=30)
X = [ (1 + vi/2 * cos(ui/2)) * cos(ui) for ui in u, vi in v ]
Y = [ (1 + vi/2 * cos(ui/2)) * sin(ui) for ui in u, vi in v ]
Z = [ vi/2 * sin(ui/2) for ui in u, vi in v ]
surface(X, Y, Z; c=:viridis, alpha=0.97, colorbar=false, title="The Mobius Strip")
Parameter sets the longitudinal displacement along the average circle of a unit radius, and — lateral displacement. A semi-argument in the term containing v, it is responsible for the gradual reversal of the line forming an angle with a full bypass.
The Battle Surface
The Battle Surface (Werner Boy is a German mathematician and physicist) this is a smooth immersion of the real projective plane into three-dimensional Euclidean space . The Battle surface provides a geometric embodiment of an undirected closed surface with a self-intersection forming a curve of triple points. The surface is the result of a search for a map of a projective plane devoid of folds and sharp edges, and allows for several analytical parameterizations.
The paper presents two implementations.
Classical parameterization of Combat uses polynomial trigonometric combinations:
Projective parameterization is based on the central projection of a sphere. First, the coordinates on the unit sphere are set:
Then the auxiliary denominator is calculated.:
And the final coordinates of the surface:
# Classic
n1 = 100
u1 = range(0, π, length=n1)
v1 = range(0, 2π, length=n1)
U1, V1 = [u_i for u_i in u1, _ in v1], [v_j for _ in u1, v_j in v1]
x1 = @. (cos(U1) * sin(2V1) + sin(U1) * cos(V1) * cos(2V1) - sin(U1) * sin(V1) * cos(2V1)) / 2
y1 = @. (sin(U1) * sin(2V1) - cos(U1) * cos(V1) * cos(2V1) + cos(U1) * sin(V1) * cos(2V1)) / 2
z1 = @. (cos(U1) * cos(2V1) + sin(U1) * cos(2V1)) / 2
p1 = surface(x1, y1, z1,
fill_z = nothing,
linewidth = 0,
legend = false,
aspect_ratio = :equal,
colorbar = false,
seriescolor = :hsv,
alpha = 0.7,
title = "Battle Surface (Classic)",
camera = (45, 30))
# The projective
n2 = 180
u2 = range(0, π, length=n2)
v2 = range(0, 2π, length=n2)
U2, V2 = [u_i for u_i in u2, _ in v2], [v_j for _ in u2, v_j in v2]
X = @. sin(U2) * cos(V2)
Y = @. sin(U2) * sin(V2)
Z = @. cos(U2)
D = @. X^2 + Y^2 + Z^2 + 1
x2 = @. sqrt(2) * X^2 / D
y2 = @. sqrt(2) * X * Y / D
z2 = @. Y / D
p2 = surface(x2, y2, z2,
fill_z = nothing,
linewidth = 0,
legend = false,
aspect_ratio = :equal,
colorbar = false,
seriescolor = :turbo,
alpha = 0.7,
title = "Battle Surface (projective)",
camera = (35, 25))
plot(p1, p2, layout=(1, 2), size=(1200, 550))
Both parameterizations generate a surface that is homeomorphic to , demonstrating various aspects of its geometry.
The Riemann surface
The Riemann surface is a natural area of uniqueness of a multivalued analytic function. In this example, the Riemann surface of a complex square root is considered. , which implements the covering of a complex plane with a branch point at zero. When traversing the origin along a closed contour, the argument changes to , which leads to a change in the sign of the root and requires switching to the second cover sheet.
The complex variable is given in the polar form:
The two leaves of the surface correspond to the two branches of the root:
r = range(0, 2, length=50)
theta = range(0, 2π, length=50)
R, Theta = [r_i for r_i in r, _ in theta], [theta_j for _ in r, theta_j in theta]
Z = R .* exp.(1im .* Theta)
W1 = sqrt.(R) .* exp.(1im .* Theta ./ 2)
W2 = -W1
X1 = real.(Z); Y1 = imag.(Z); Z1 = real.(W1)
X2 = real.(Z); Y2 = imag.(Z); Z2 = real.(W2)
p = plot(xlabel = "Re(z)", ylabel = "Im(z)", zlabel = "Re(w)",
title = "The Riemann surface", camera = (-45, 45), aspect_ratio = :equal)
surface!(p, X1, Y1, Z1, alpha=0.9, colorbar=false, fillcolor=:cool, linewidth=0)
surface!(p, X2, Y2, Z2, alpha=0.9, colorbar=false, fillcolor=:cool, linewidth=0)
display(p)
In three-dimensional representation along the axes and the real and imaginary parts are postponed , and on the axis — the actual part of the corresponding value . The joint construction of two sheets intersecting along the branching ray gives a complete picture of the topology of the cover.
The crossed hood
A crossed cap is a standard model of an undirected closed surface equivalent to a projective plane. with a Mobius sheet glued along the boundary circle. Unlike the surface of the Battlefield, the crossed hood contains a line of special points along which the surface self-intersects. Geometrically, this model can be represented as a sphere in which diametrically opposite points of the equatorial circle are identified, which generates a characteristic feature in the form of intersecting "tongues".
Parametric description:
n = 100
u = range(0, π, length=n)
v = range(0, π, length=n)
U, V = [u_i for u_i in u, _ in v], [v_j for _ in u, v_j in v]
x = @. sin(2U) * sin(V)^2
y = @. sin(U) * sin(2V)
z = @. cos(U) * sin(V)^2 - sin(U)^2 * cos(V)
surface(x, y, z,
fill_z = nothing,
linewidth = 0,
legend = false,
aspect_ratio = :equal,
colorbar = false,
title = "The crossed hood",
seriescolor = :turbo,
alpha = 0.8)
Parameters and correspond to the angular coordinates on the original sphere. Combinations of sines and cosines determine the immersion of a sphere with an antipodal involution into three-dimensional space.
Conclusion
The examples discussed illustrate the fundamental difference between local geometry and global topology of surfaces. The Mobius strip, the Combat surface, and the crossed hood demonstrate three different ways of realizing non-orientability in three-dimensional space: from a one-sided surface with a boundary to closed models of a projective plane with self-intersections of various structures. The Riemann surface, in turn, reveals the connection between topology and complex analysis, showing how the ambiguity of analytic functions is naturally resolved by passing to covers.
These computational models can serve as a basis for studying more complex topological invariants, such as homotopy groups, Euler classes, and obstacles to orientability of vector bundles. From an applied perspective, understanding the structural topology of undirected manifolds finds application in liquid crystal theory, condensed matter physics, and modern cosmology in analyzing possible global structures of space-time.