Origin, Development, and Dissemination of Differential Geometry in Mathema

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1 Origin, Development, and Dissemination of Differential Geometry in Mathematical History The Borough of Manhattan Community College -The City University of New York Fall 2016 Meeting of the Americas Section of International Study Group on Relations between History and Pedagogy of Mathematics HPM-Americas Point Loma Nazarene University San Diego, California, U.S.A. October 15 16, 2016

2 Abstract Origin of Geometry Applying Geometry in daily life dated back to ancient Egyptian period. At approximately 3, 000 BC, Egyptians began using Geometry to solve many difficult problems such as surveying of land, construction of buildings, and Astronomy related to Earth Measurement. Ancient Geometry was used during early Egyptian, Greek, Indiana, and Chinese civilization.

3 Abstract Time Line of Geometry Development From 9th century thru 10th century, the use of geometry was popular in Islamic Art Design. From 11th century thru 15th century, Sacred Geometry was established as a special geometry in Religious Art. From 14th century thru 17th century, Geometry was generally used in Renaissance Western Art. From 18th century thru 19th century, the creation of Non-Euclidean Geometry became the origin of Modern Geometry.

4 Abstract Differential Geometry Origin of Differential Geometry began with geometric properties of curves and surfaces in space at the beginning of the 19th century. The ancient study was related to Calculus techniques, Carl Friedrich Gauss on Gauss Curvature, Bernhard Riemann on Riemannian Manifold in 1854 and Albert Einstein s General Relativity Theory between 1907 and The contemporary study focused on geometric structures of various manifolds including non-euclidean spaces. It was enhanced by Shiing-Shen Chern on Chern s Characteristic Classes, Shing-Tung Yau on Calabi-Yau Manifold and Minimal Surfaces Theory, and Henry Wente on Wente Torus as the Constant-Mean-Curvature Surface, etc.

5 Abstract Connections between Differential Geometry and Other Mathematical Fields The growing study of Geometry has contributed to the expending development in geometric-related mathematical fields. For examples, the creation of Differential geometry came from the connection between Geometry and Calculus. The origin of Algebraic Geometry was the connection between Geometry and Algebra. The origin of Geometric Analysis was the connection between Geometry and Analysis.

6 Overview A The Origin of Geometry B The Ancient Study of Differential Geometry C The Contemporary Development of Differential Geometry D Mathematicians and Their Contributions to Differential Geometry E Geometric-Related Mathematical Fields

7 A-Origin of Geometry The origin of Geometry is motivated by solving problems in Astronomy. Geometry is also called Earth Measurement. Geometry comes from Ancient Greek, geo-earth; metry-measurement. Earth Measurement is concerned with the shape, size, figure, mapping the positions of stars and planets in space, spatial relationships among various moving objects in space, and properties of surrounding space.?? Shapes of Space Objects-surface?? Orbits of moving objects in space-space curve

8 B-The Ancient Study of Differential Geometry Differential Geometry arises from studying the geometric properties of curves and surfaces in space as the main topics during 18th century and 19th century. Differential Geometry was founded by Gaspard Monge and C.F. Gauss at the beginning of 19th century. The important contribution were made by many mathematicians in 19th century such as B. Riemannian, E.B. Christoffel, and C. G. Ricci. The work was collected and systematized at the end of 19th century by J.G.Darboux and Luigi Bianchi.

9 B-Differential Geometry and Calculus Differential Geometry studies geometric properties of objects by applying the techniques of Calculus. Calculus was invented in the late 17th century by Sir Isaac Newton and Gottfried Wilhelm Von Leibniz. 1 Use Infinitesimals to describe non-zero quantities with infinitely small amount Invent the idea of Limit and Pointwise. 2 Invent the definition of Derivative and Integral formulated in terms of limits. The idea of calculating pointwise curvature plays an important role in Differential Geometry. The concept of curvature was developed in 18th century as an effective and efficient measurement tool to estimate the bending degree of a curve or surface:?? How much is a given curve or a surface deviates from being a straight line or a plane

10 B-Curvature as Degree of Turning Angles, Bent Curved, Curved Surfaces, and Non-Flat Space Differential Geometry studies geometric properties of objects by applying the techniques of Calculus. Computing the curvature K of a curve : For example, a circle with a radius r : κ = 1 r (in Calculus); or a curve with the curvature value varying from point to point Calculating the curvature of a surface: Carl Friedrich Gauss is well-known for his work of Gauss Curvature on the theory of 2-dim surfaces in 3-dim space. Gauss Curvature as an important method of measuring the pointwise curvature of a surface is the product of the greatest and least curvatures of all curves passing through that point on the surface. Calculating the curvature of a space

11 B-Introduction Various Curvatures 1 Sectional Curvature Sec (called Gauss Curvature): Sec depends on a plane σ p spanned by two independent tangent vectors (α, β) in the tangent space T p at one point p on a surface S. 2 Ricci Curvature Ric: For a tangent vector ξ, Ric(ξ) denotes the trace of sectional curvatures taken over all of possible 2-dim tangent plane σ p containing ξ. 3 Principle Curvature κ: Principle curvature depends on a point p on a surface S. For a normal plane N p spanned by a normal vector ν and a tangent vector η, we consider the curve C between N p and a surface S Principle curvature κ at a point p denotes the max κ 1 and min κ 2 of the curvature of C. 4 Mean Curvature: H is the average of the principle curvatures.

12 B-Differential Geometry and Physics by Albert Einstein s Theory of General Relativity Einstein Equations ( )E = mc 2 where energy and mass are equivalent and transmutable. R ij 1 2 R g ij = 8πT ij where R ij is the Ricci Curvature and g ij is the metric. In the Theory of General Relativity, Einstein imagined the space-time as a manifold (a space equipped with coordinates and metric structures). A space-event is considered as a point. The world line of the particle is traced by a curve. The curvature on this space-time manifold is determined by the distribution of energy and matter. In universe, a star or a black hole can be described as a curved or bent space-time manifold. Physicists believes that the curvature is related to the gravitational field of a star according to Einsteins Equation.

13 B-- The Ancient Study of Differential Geometry 1 On a manifold M (a space equipped with a metric structure) 2 Coordinate at a point 3 Tangent Plane at a point 4 Tangent Vectors on the tangent plane 5 Metric, Metric Field A metric field is an inner Product for the tangent plane at each point and defined as a symmetric positive-definite matrix. 6 Connection (Levi-Civita connection) to connect vectors at different points 7 Orthonormal frame field 8 Curvatures 9 Geodesic and minimizing problems problems of the shortest curve connecting two given points the minimizing area problems of surfaces with the fixed boundary the energy-minimizing problems of maps between two manifolds

14 B-Riemannian Manifold and Riemannian Geometry Riemannian Geometry is the branch of differential geometry that studies a Riemannian manifold M (also called smooth manifolds) with a Riemannian metric g. The metric g at each point p is an inner product on the tangent space T p M that varies smoothly from point to point. (M, g) makes it possible to calculate the geometric properties such as angle between vectors, length of curves, area and volume, and gradient of function and divergence of vector fields. Riemannian Geometry was originated with the vision of Bernhard Riemann (one of Gausss students) in The technology on Riemannian manifolds can be applied to the study of any differentiable manifolds with higher dimensions. Bernhard Riemanns was not received its proper attention in the mathematical society until Einsteins equation involved with Ricci curvatures on the Riemannian manifold in the theory of General Relativity (in 1915).

15 C-The Contemporary Development of Differential Geometry Since the 19th century, Differential Geometry has grown into a field concerned more generally with the geometric structures on differential manifolds. Riemannian Manifold and Riemannian Geometry Finsler Manifold and Finsler Geometry Complex- Kahler Manifold and Complex-Kahler Geometry Calabi-Yau Manifold?? For a physical space, which metric construction will best fit the physical space to study the geometric properties

16 D -Mathematicians and Their Contributions

17 D-Innovator of Differential Geometry in the 20th Century: Shiing-Shen Chern Chern s Contribution to Geometry 1 Geometric Structures and Their Equivalence Problems 2 Integral Geometry 3 Euclidean Differential Geometry 4 Minimal Surfaces and Minimal Submanifolds 5 The Generalized Gauss-Bonnet Theorem 6 Chern s Characteristic Classes 7 Chern and Yang-Mills Connection Mathematical Research Institutes established by Chern 1 The Mathematical Science Research Institute in Berkeley, California, USA (MSRI) The Mathematical Institute of Academia Sinica in TaiWan The Nankai Institute for Mathematics in Tianjin, China-1985

18 D-Significant Contributor: Shing-Tung Tau (Field Medal in 1982) 1 Solving Conjectures Calabi Conjecture Positive mass conjecture and existence of black holes Smith Conjecture Hermitian Yang-Mills connection and stable vector bundles Frankel Conjecture Mirror Conjecture 2 Creating new methods and concepts Gradient estimates and Harnack inequalities Uniformization of complex manifolds Harmonic maps and rigidity Minimal submanifolds 3 Establishing Mathematical Research Institutes Institute of Mathematical Sciences at Chinese University of Hong Kong The Morningside Center of mathematics of The Chinese Academy of Sciences

19 D-Shihshu Wei s Contribution to p-harmonic Geometry Professor Shihshu Walter Wei is an expert in the field of p-harmonic Geometry. Shihshu Walter Wei s Contribution to p-harmonic Geometry 1 Studying p-harmonic functions, p-harmonic maps, p-harmonic differential forms, p-harmonic operators 2 Discovering the second variational formulas for p-harmonic maps 3 Defining p-ssu manifolds and p SSU index, 4 Working on p-energy minimizing problem and Stability, 5 Creating p-balanced growth estimates, 6 Discovering homotopy groups represented by p-harmonic maps, 7 Finding p-harmonic approach to the Generalized Bernstein Problem, 8 Finding p-harmonic approach to minimal surfaces, 9 Generalizing the Bochners Method in p-harmonic point of view, 10 Studying p-harmonic inequalities on manifolds, 11 Studying F -Harmonic maps extended from the p-harmonic map.

20 D-Henry Wente s Contribution to Constant-Mean-Curvature Surfaces The study of CMC surfaces is one of the main topics in Differential Geometry. CMC are surfaces with constant mean curvatures. Professor Henry Wente is known for his 1984 discovery of the CMC surface of Wente Torus. Wente Torus is a counter-example of Hopf s Conjecture. Henry Wente disproved a conjecture of Heinz Hopf. Hopf Conjecture: Every closed compact constant-mean-curvature surface is a sphere.

21 E-Geometric-Related Mathematical Fields The trend of technology of studying Differential Geometry is to combine the techniques in various fields such as Algebra, Real and Complex Analysis, Topology, Partial Differential Equations (PDE) 1 Differential Geometry and Algebra Transformation under changes of coordinates system on manifold s plays a similar role as the study of Algebra under the action in the group theory. Algebraic Geometry is a discipline at the interface of Differential Geometry and Algebra. 2 Differential Geometry and Topology The relationship between some the local behavior (estimated by curvature conditions) and the global topological properties of the manifold. 3 Differential Geometry and PDE The study of p-harmonic maps or p-harmonic differential forms between manifolds equipped with various metric structures will be involved with solving the non-linear partial differential equations. Geometric Analysis is a discipline at the interface of Differential Geometry and Partial Differential Equations.

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