Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology. (Wikipedia).
Riemannian Geometry - Definition: Oxford Mathematics 4th Year Student Lecture
Riemannian Geometry is the study of curved spaces. It is a powerful tool for taking local information to deduce global results, with applications across diverse areas including topology, group theory, analysis, general relativity and string theory. In these two introductory lectures
From playlist Oxford Mathematics Student Lectures - Riemannian Geometry
MATH331: Riemann Surfaces - part 1
We define what a Riemann Surface is. We show that PP^1 is a Riemann surface an then interpret our crazy looking conditions from a previous video about "holomorphicity at infinity" as coming from the definition of a Riemann Surface.
From playlist The Riemann Sphere
Riemannian Geometry - Examples, pullback: Oxford Mathematics 4th Year Student Lecture
Riemannian Geometry is the study of curved spaces. It is a powerful tool for taking local information to deduce global results, with applications across diverse areas including topology, group theory, analysis, general relativity and string theory. In these two introductory lectures
From playlist Oxford Mathematics Student Lectures - Riemannian Geometry
Understanding and computing the Riemann zeta function
In this video I explain Riemann's zeta function and the Riemann hypothesis. I also implement and algorithm to compute the return values - here's the Python script:https://gist.github.com/Nikolaj-K/996dba1ff1045d767b10d4d07b1b032f
From playlist Programming
Riemann Sum Defined w/ 2 Limit of Sums Examples Calculus 1
I show how the Definition of Area of a Plane is a special case of the Riemann Sum. When finding the area of a plane bound by a function and an axis on a closed interval, the width of the partitions (probably rectangles) does not have to be equal. I work through two examples that are rela
From playlist Calculus
Curvature of a Riemannian Manifold | Riemannian Geometry
In this lecture, we define the exponential mapping, the Riemannian curvature tensor, Ricci curvature tensor, and scalar curvature. The focus is on an intuitive explanation of the curvature tensors. The curvature tensor of a Riemannian metric is a very large stumbling block for many student
From playlist All Videos
This talk is about some properties of plane curves used in the Riemann-Roch theorem. We first show that every nonsingular curve is isomorphic to a resolution of a plane curve with no singularities worse than ordinary double points (nodes). We then calculate the genus of plane curves with o
From playlist Algebraic geometry: extra topics
Riemann Roch: structure of genus 1 curves
This talk is about the Riemann Roch theorem in the spacial case of genus 1 curves or Riemann surface. We show that a compact Riemann surface satisfying the Riemann Roch theorem for g=1 is isomorphic to a nonsingular plane cubic. We show that this is topologically a torus, and use this to s
From playlist Algebraic geometry: extra topics
What is the Riemann Hypothesis?
This video provides a basic introduction to the Riemann Hypothesis based on the the superb book 'Prime Obsession' by John Derbyshire. Along the way I look at convergent and divergent series, Euler's famous solution to the Basel problem, and the Riemann-Zeta function. Analytic continuation
From playlist Mathematics
Geometry of Surfaces - Topological Surfaces Lecture 1 : Oxford Mathematics 3rd Year Student Lecture
This is the first of four lectures from Dominic Joyce's 3rd Year Geometry of Surfaces course. The four lectures cover topological surfaces and conclude with a big result, namely the classification of surfaces. This lecture provides an introduction to the course and to topological surfaces.
From playlist Oxford Mathematics Student Lectures - Geometry of Surfaces
Pierre Pansu: Differential forms and the Hölder equivalence problem - Part 1
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Geometry
I give a proof of the Cartan-Hadamard theorem on non-positively curved complete Riemannian manifolds. For more details see Chapter 7 of do Carmo's "Riemannian geomety". If you find any typos or mistakes, please point them out in the comments.
From playlist Differential geometry
Crash course in Riemanian geometry(Lecture – 02) by Harish Seshadri
Geometry, Groups and Dynamics (GGD) - 2017 DATE: 06 November 2017 to 24 November 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru The program focuses on geometry, dynamical systems and group actions. Topics are chosen to cover the modern aspects of these areas in which research has b
From playlist Geometry, Groups and Dynamics (GGD) - 2017
Ludovic Rifford: Geometric control and sub-Riemannian geodesics - Part I
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Geometry
Jialong Deng - Enlargeable Length-structures and Scalar Curvatures
38th Annual Geometric Topology Workshop (Online), June 15-17, 2021 Jialong Deng, University of Goettingen Title: Enlargeable Length-structures and Scalar Curvatures Abstract: We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a
From playlist 38th Annual Geometric Topology Workshop (Online), June 15-17, 2021
[BOURBAKI 2019] Higher rank Teichmüller theories - Pozzetti - 30/03/19
Beatrice POZZETTI Higher rank Teichmüller theories Let Γ be the fundamental group of a compact surface S with negative Euler characteristic, and G denote PSL(2, R), the group of isometries of the hyperbolic plane. Goldman observed that the Teichmüller space, the parameter space of marked
From playlist BOURBAKI - 2019
Some identities involving the Riemann-Zeta function.
After introducing the Riemann-Zeta function we derive a generating function for its values at positive even integers. This generating function is used to prove two sum identities. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist The Riemann Zeta Function