Theorems in Riemannian geometry | Riemannian geometry | Theorems in topology
In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. If M is a complete, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval then M is homeomorphic to the n-sphere. (To be precise, we mean the sectional curvature of every tangent 2-plane at each point must lie in .) Another way of stating the result is that if M is not homeomorphic to the sphere, then it is impossible to put a metric on M with quarter-pinched curvature. Note that the conclusion is false if the sectional curvatures are allowed to take values in the closed interval . The standard counterexample is complex projective space with the Fubini–Study metric; sectional curvatures of this metric take on values between 1 and 4, with endpoints included. Other counterexamples may be found among the rank one symmetric spaces. (Wikipedia).
The video explains how to determine the center and radius of a sphere. http://mathispower4u.yolasite.com/
From playlist Vectors
Finding the volume and the surface area of a sphere
👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo
From playlist Volume and Surface Area
From playlist Drawing a sphere
Learn how to determine the volume of a sphere
👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo
From playlist Volume and Surface Area
How do you find the surface area of a sphere
👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo
From playlist Volume and Surface Area
Find the volume of a sphere given the circumference
👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo
From playlist Volume and Surface Area
Equation of Sphere given Endpoints of Diameter
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Equation of Sphere given Endpoints of Diameter
From playlist Calculus
How do you find the volume of a sphere
👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo
From playlist Volume and Surface Area
Given the circumference how do you find the surface area of a hemisphere
👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo
From playlist Volume and Surface Area
Johnathan Bush (7/8/2020): Borsuk–Ulam theorems for maps into higher-dimensional codomains
Title: Borsuk–Ulam theorems for maps into higher-dimensional codomains Abstract: I will describe Borsuk-Ulam theorems for maps of spheres into higher-dimensional codomains. Given a continuous map from a sphere to Euclidean space, we say the map is odd if it respects the standard antipodal
From playlist AATRN 2020
Extended Gauss' Theorem | MIT 18.02SC Multivariable Calculus, Fall 2010
Extended Gauss' Theorem Instructor: Joel Lewis View the complete course: http://ocw.mit.edu/18-02SCF10 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 18.02SC: Homework Help for Multivariable Calculus
Benson Farb, Part 3: Reconstruction problems in geometry and topology
29th Workshop in Geometric Topology, Oregon State University, June 30, 2012
From playlist Benson Farb: 29th Workshop in Geometric Topology
My Favorite Theorem: The Borsuk-Ulam Theorem
Many thanks for 10k subscribers! Fun video for you from Topology: The Borsuk-Ulam Theorem. One interpretation of this is that on the surface of the earth, there must be some point where it and its antipode (the spot exactly opposite it) have the exact same temperature and pressure. More ge
From playlist Cool Math Series
C. Sormani - Intrinsic Flat and Gromov-Hausdorff Convergence 2
We introduce various notions of convergence of Riemannian manifolds and metric spaces. We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with uniform lower bounds on their scalar curvature. We close the course by presenting methods and the
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
Žiga Virk (4/24/21): A counter-example to Hausmann's conjecture
Title: A counter-example to Hausmann's conjecture Abstract: In 1995 Jean-Claude Hausmann proved that a closed compact Riemannian manifold is homotopy equivalent to its Vietoris-Rips complex for small values of the scale parameter. He then conjectured that the connectivity of Vietoris-Rip
From playlist Vietoris-Rips Seminar
Topology is weird: The Ham Sandwich Theorem
Keep exploring at ► https://brilliant.org/TreforBazett. Get started for free, and hurry—the first 200 people get 20% off an annual premium subscription. Today we talk about my favourite math theorem: the Ham Sandwich theorem. Consider a sandwich with three components. Then the theorem cla
From playlist Cool Math Series
C. Sormani - Intrinsic Flat and Gromov-Hausdorff Convergence 2 (version temporaire)
We introduce various notions of convergence of Riemannian manifolds and metric spaces. We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with uniform lower bounds on their scalar curvature. We close the course by presenting methods and the
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
Connections between classical and motivic stable homotopy theory - Marc Levine
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From playlist Mathematics
This lecture was held by Abel Laureate John Milnor at The University of Oslo, May 25, 2011 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations. Program for the Abel Lectures 2011 1. "Spheres" by Abel Laureate John Milnor, Institute for Mathematical
From playlist Abel Lectures
Multivariable Calculus | The equation of a sphere.
We derive the equation of a sphere in R^3 and look at some examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Common Surfaces in Multivariable Calculus