PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Topology → Differential Topology ↓


Smooth Manifolds


In the world of differential topology, a "smooth manifold" is a fundamental concept that combines ideas from many areas of mathematics, including calculus, topology, and geometry. A manifold is a geometric object that looks like a Euclidean space around each point. When we say that a manifold is "smooth," we mean that it has a structure that allows us to perform calculus operations — such as differentiation — smoothly on its points.

Let us take this slowly and first consider what it means for something to locally look like Euclidean space.

What is a manifold?

A manifold is a space that resembles Euclidean space of a specific dimension on a small scale. For example, consider a surface like the surface of the Earth. Locally, or if you only look at a small region, it seems flat - like a plane, which is the 2-dimensional Euclidean space we represent as R 2. However, it is actually a part of something much larger and curved, namely a globe.

Technically, a manifold is a topological space that is locally Euclidean. This means that every point has a neighborhood that is homeomorphic (i.e., topologically identical) to an open set in R n. Depending on what the dimension n is, we have a 1-manifold, such as a circle; a 2-manifold, such as the surface of a sphere; etc.

Charts and atlases

To formally define manifolds, mathematicians use the concepts of "chart" and "atlas".

  • Chart: A chart on a manifold is a pair (U, φ) where U is an open subset of the manifold, and φ (phi) is a homeomorphism from U to an open subset of R n.
  • Atlas: A collection of charts that covers the entire manifold. This means that every point of the manifold is in the domain of at least one chart of the atlas.

The concept of an atlas is similar to using multiple maps to cover a wide area, such as a city. Each map covers a different part, just like each chart in an atlas covers a part of a manifold.

Lubrication

The smoothness condition on manifolds deals with how these charts transition or overlap with each other. When transitioning from one chart to another, the change of coordinates must be smooth - infinitely differentiable. Such transitions are characterized by diffeomorphisms.

Diffeomorphism: A function between smooth manifolds that is smooth, one-to-one, surjective, and has a smooth inverse.

The requirement of diffusivity for changes in coordinates ensures that there are no "jumps" or "cracks" in the manifold, maintaining a seamless global structure.

Examples of smooth manifolds

Circle

Consider the simplest smooth manifold: a 1-dimensional sphere, also called a circle. Here is a representation of the circle:

    x 2 + y 2 = 1
x 2 + y 2 = 1

Sphere

A two-dimensional sphere, or simply sphere, is another example:

    x 2 + y 2 + z 2 = 1
x 2 + y 2 + z 2 = 1

Locally, a sphere looks like a plane, making it a 2-manifold. Globally, it is a closed, compact surface with no boundary.

Torus

The torus, or donut shape, is a classic example of a manifold that is easy to understand visually:

Unlike a circle or sphere which have no holes, a torus has a hole:

Tangent space

In a smooth manifold, like a curved surface, there is a notion of a tangent plane or tangent space at each point. The tangent space consists of the tangent vectors at a point and provides a linear approximation to the manifold around that point.

Think about how a tangent line to a curve touches only one point on the curve. Likewise, the tangent space will touch only one point on the manifold.

More complex examples and concepts

Mobius strip

Consider the Möbius strip, which is a non-orientable surface, with only one side and one boundary component.

Representing the Möbius strip mathematically requires more advanced concepts, but it serves as an important manifold in a variety of applications.

Poincaré conjecture and Ricci flow

The Poincaré conjecture - which was famously proven in the early 21st century - deals with manifolds and shows the importance of understanding these structures in depth. The solution involves understanding the topology of 3-manifolds and uses methods such as Ricci Flow to classify 3-dimensional spaces - a process akin to "flattening" a manifold.

Applications of smooth manifolds

The applications of smooth manifolds extend far beyond mathematics into fields such as physics, engineering, and computer science. For example:

  • In physics, smooth manifolds form the basis of general relativity, where spacetime is modeled as a 4-dimensional smooth manifold.
  • In robotics, the configuration space of a robot is a manifold that represents possible positions and orientations.
  • In computer graphics, rendering realistic surfaces involves understanding the local and global properties of structures such as manifolds.

Mathematical representation

Mathematica or other computational packages can use manifolds defined via parametric equations or differential equations to automate certain calculations or to model real-world systems:

    f(x, y) = z (such smooth functions create level sets that are manifolds)
f(x, y) = z (such smooth functions create level sets that are manifolds)

These representations show how functions define the structure and smoothness properties of manifolds and help advance areas where these concepts are fully applicable.

Conclusion

In short, smooth manifolds are practical abstractions that help encapsulate and generalize the idea of surfaces or more complex geometries. By accommodating a broader structure while retaining the local simplicity of Euclidean space, they allow mathematicians and scientists to explore the intricacies of various fields with greater flexibility and rigor.

As you progress in your understanding of smooth manifolds, remember that they are not just a theoretical concept, but a doorway to many practical and theoretical applications spanning modern mathematics and beyond.


PHD → 3.3.1


U
username
0%
completed in PHD

← Prev (3.3)
Differential Topology
Next (3.3.2) →
Tangent Spaces

Comments 



Smooth Manifolds

https://www.buddymath.com/phd/3.3.1?bmal=en