Theorems in Riemannian geometry

Rauch comparison theorem

In Riemannian geometry, the Rauch comparison theorem, named after Harry Rauch, who proved it in 1951, is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, it states that for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread. The statement of the theorem involves two Riemannian manifolds, and allows to compare the infinitesimal rate at which geodesics spread apart in the two manifolds, provided that their curvature can be compared. Most of the time, one of the two manifolds is a "comparison model", generally a manifold with constant curvature , and the second one is the manifold under study : a bound (either lower or upper) on its sectional curvature is then needed in order to apply Rauch comparison theorem. (Wikipedia).

Video thumbnail

Learning to evaluate the sum of two angles in radians, tan

πŸ‘‰ Learn how to evaluate the tangent of an angle in degrees using the sum/difference formulas. To do this, we first express the given angle as a sum or a difference of two (easy to evaluate) angles, then we use the unit circle and the Pythagoras theorem to identify the angles and obtain all

From playlist Sum and Difference Formulas

Video thumbnail

Using the addition of two angles formula and sine

πŸ‘‰ Learn how to evaluate the sine of an angle in radians using the sum/difference formulas. To do this, we first express the given angle as a sum or a difference of two (easy to evaluate) angles, then we use the unit circle and the Pythagoras theorem to identify the angles and obtain all th

From playlist Sum and Difference Formulas

Video thumbnail

D-Day - The German Counterattack 1944

Find out what happened when the Germans counterattacked the Allied landing beaches on 6 June 1944, with surprising and potentially devastating results. Enjoy reading, then check out my latest book, The Bridge Busters: The First Dambusters and the Race to Save Britain https://www.amazon.c

From playlist Battle of Normandy 1944

Video thumbnail

Evaluate the difference of two angles for sine

πŸ‘‰ Learn how to evaluate the sine of an angle in radians using the sum/difference formulas. To do this, we first express the given angle as a sum or a difference of two (easy to evaluate) angles, then we use the unit circle and the Pythagoras theorem to identify the angles and obtain all th

From playlist Sum and Difference Formulas

Video thumbnail

Jeffrey Achter, Equidistribution counts abelian varieties

VaNTAGe Seminar, February 22, 2022 License: CC-BY-NC-SA Links to some of the papers mentioned in this talk are listed below. Sutherland: https://arxiv.org/abs/1604.01256 Gekeler: https://academic.oup.com/imrn/article/2003/37/1999/863196 Job Rauch: https://www.universiteitleiden.nl/binar

From playlist Curves and abelian varieties over finite fields

Video thumbnail

Determining the sine of the sum of two angles

πŸ‘‰ Learn how to evaluate the sine of an angle in radians using the sum/difference formulas. To do this, we first express the given angle as a sum or a difference of two (easy to evaluate) angles, then we use the unit circle and the Pythagoras theorem to identify the angles and obtain all th

From playlist Sum and Difference Formulas

Video thumbnail

Einstein's Quantum Riddle | Full Documentary | NOVA | PBS

Join scientists as they grab light from across the universe to prove quantum entanglement is real. #NOVAPBS Official Website: https://to.pbs.org/3vqiMpg Einstein called it β€œspooky action at a distance,” but today quantum entanglement is poised to revolutionize technology from computers t

From playlist Full episodes I NOVA

Video thumbnail

Using sum of two angles to evaluate an angle in radians for tangent, tan

πŸ‘‰ Learn how to evaluate the tangent of an angle in degrees using the sum/difference formulas. To do this, we first express the given angle as a sum or a difference of two (easy to evaluate) angles, then we use the unit circle and the Pythagoras theorem to identify the angles and obtain all

From playlist Sum and Difference Formulas

Video thumbnail

Using difference of two angles with tangent to evaluate

πŸ‘‰ Learn how to evaluate the tangent of an angle in degrees using the sum/difference formulas. To do this, we first express the given angle as a sum or a difference of two (easy to evaluate) angles, then we use the unit circle and the Pythagoras theorem to identify the angles and obtain all

From playlist Sum and Difference Formulas

Video thumbnail

Harsha Hutridurga: A new approach to study strong advection problems

The lecture was held within the framework of the Hausdorff Trimester Program Multiscale Problems: Workshop on Numerical Inverse and Stochastic Homogenization. (16.02.2017) In this talk, I shall be attempting to give an overview of a new weak convergence type tool developed by myself, Thom

From playlist HIM Lectures: Trimester Program "Multiscale Problems"

Video thumbnail

Radian Measure (Mini Lesson) - Algebra 2

http://www.youtube.com/vinteachesmath This video provides a mini lesson on the concept of radian measure. In particular, this video shows how the unit circle, circumference, and degree measure of an angle can be used to explain the concept of radian measure. This video is appropriate fo

From playlist Trigonometry (old videos)

Video thumbnail

ExtremwetterschΓ€den: Wer trΓ€gt die Verantwortung?

Max-Planck-Forum in Kooperation mit dem Max-Planck-Institut fΓΌr Sozialrecht und Sozialpolitik Aufzeichnung vom 15.11.2022 Auf dem Podium: Prof. Dr. Ulrich Becker, Direktor am Max-Planck-Institut fΓΌr Sozialrecht und Sozialpolitik Ernst Rauch, Chief Climate and Geo Scientist, Munich Re Dr

From playlist Videos auf Deutsch

Video thumbnail

How to evaluate using the sum of two angles for sine

πŸ‘‰ Learn how to evaluate the sine of an angle in radians using the sum/difference formulas. To do this, we first express the given angle as a sum or a difference of two (easy to evaluate) angles, then we use the unit circle and the Pythagoras theorem to identify the angles and obtain all th

From playlist Sum and Difference Formulas

Video thumbnail

Pre-Calculus - Using the difference of two angles to evaluate the an angle for cosine, cos195

πŸ‘‰ Learn how to evaluate the cosine of an angle in radians using the sum/difference formulas. To do this, we first express the given angle as a sum or a difference of two (easy to evaluate) angles, then we use the unit circle and the Pythagoras theorem to identify the angles and obtain all

From playlist Sum and Difference Formulas

Video thumbnail

The Geometry of Hilbert's 13th problem - Jesse Wolfson

Special Seminar on Hilbert's 13th Problem I Topic: The Geometry of Hilbert's 13th problem Speaker: Jesse Wolfson Affiliation: University of California, Irvine Date: December 5, 2019 For more video please visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

Alexandra SHLAPENTOKH - Defining Valuation Rings and Other Definability Problems in Number Theory

We discuss questions concerning first-order and existential definability over number fields and function fields in the language of rings and its extensions. In particular, we consider the problem of defining valuations rings over finite and infinite algebraic extensions

From playlist Mathematics is a long conversation: a celebration of Barry Mazur

Video thumbnail

GCSE Science Revision Chemistry "NPK Fertilisers" (Triple)

Find my revision workbooks here: https://www.freesciencelessons.co.uk/workbooks In this video, we look at how NPK fertilisers are produced. Deliberate Thought by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/) Source:

From playlist 9-1 GCSE Chemistry Paper 2 Resources

Video thumbnail

Using sum and difference formula to find the exact value with cosine

πŸ‘‰ Learn how to evaluate the cosine of an angle in radians using the sum/difference formulas. To do this, we first express the given angle as a sum or a difference of two (easy to evaluate) angles, then we use the unit circle and the Pythagoras theorem to identify the angles and obtain all

From playlist Sum and Difference Formulas

Video thumbnail

The Most Efficient Way to Destroy the Universe – False Vacuum

What if there is a way to destroy the universe so fundamentally that life as we know it will be impossible forever? OUR CHANNELS β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€ German Channel: https://kgs.link/youtubeDE Spanish Channel: https://kgs.link/youtubeES HOW CAN YOU SUPPORT US? β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€β–€

From playlist The Existential Crisis Playlist

Related pages

Toponogov's theorem | Jacobi field | Manfredo do Carmo | Conjugate points | Geodesic | Constant curvature | Ricci curvature | Sectional curvature | Riemannian manifold | Riemannian geometry | Hessian matrix | Harry Rauch