"The horizon is simply the vanishing line of perspective from your point of view, not the alleged curvature of the earth."
"Space has a kind of warp, distorted, curved look; the world around you seems to be curving in around you when your speed approaches the speed of light."
"Oceans are level even though they exhibit curvature, and this is how all the water molecules sit as close to the center of the Earth as geometrically possible."
"The uncertainty principle is a story of essentially curvature in this multi-way graph space."
"Euclidean geometry is a specific example of a way that geometry could be. But there's other options where there is curvature."
"When we visualize curved spaces or surfaces, we inevitably do it as a two-dimensional surface being bent, embedded in three-dimensional space that we live in. That is entirely a feature of our visualization capacities, not a feature of nature."
"The universe might actually be smaller than that, and the furthest galaxies that we're seeing might actually be images of our own galaxy in other words, light has traveled along a curvature for such a long time that it loops back onto itself like a sphere."
"Inflation stretches the fabric of the universe, making any existing curvature undetectable."
"To indicate that it has positive spatial curvature, we sometimes write that it has k equal to +1."
"Space actually is curved. Non-Euclidean geometry was already there for Einstein to use."
"The reason why it is curved is because the moon and the sun, to some extent, are curving it."
"The Riemann tensor tells us how initially parallel lines move apart, twist, or converge."
"The curvature scalar tells you the overall amount of curvature at every point in space-time."
"I want to imply that that's actually curving back in space."
"If you double the size of a sphere, its curvature changes."
"When the curvature of space becomes important, it's kind of like the Earth. When does the curvature of the Earth become important? When you're studying regions of the Earth which are of the order of magnitude the radius of the Earth."
"All fingerboards are slightly rounded or have a radius to give the natural curvature of your hand easier access to the strings."
"The K stands for curvature curvature equals 0 that's flat space K equals 1 that's positively curved space and it's the analog of a sphere but of course we are not talking about a 2-dimensional sphere space is not a two-dimensional sphere it could be a 3-dimensional sphere."
"...if K is positive, that's the sphere case... if it's flat, then it's as if every galaxy was exactly at the escape velocity... if K is negative, that corresponds to being above the escape velocity."
"If you ever have a positive manifold, a positive scalar curvature, Richie flow is going to send it to become singular in finite time."
"The concept of space-time curvature is essential in understanding how black holes interact with their surroundings, highlighting the ongoing challenges faced by physicists."
"So this curvature of the space really is a relativistic effect."
"Gravity does not cause the curvature of space-time; it is the curvature of space-time."
"The equations describe the curvature of space-time by treating it as being flat at infinitesimally small distances."
"The Ricci tensor is telling us about the curvature of space."
"I love the tumble home on this car, in other words, the curvature from the midsection of the body up toward the roof."
"The Fifth Dimension is curved in a way we can't see it."
"The mathematics of the saddle surface is such that it has uniform curvature everywhere, and it's negatively curved."
"Curvature causes initially parallel trajectories to become non-parallel."
"It's lovely and curved, quite deep."
"A sail is made out of flat panels; these flat panels get a curve to each other by bending the glue area."
"If we're going to tackle this problem of 'is can space-time be curved' we have to approach it intrinsically."
"Is there a way for this little guy to determine even though they're confined to that space and they're not allowed to go to a higher dimension, can they tell that they're in a curved space? The answer is yes."
"This guy here is what we call the Riemann curvature tensor."
"This is the end-all be-all for determining if your space is curved or not."
"Sectional curvature measured how neighboring geodesics either converged together or diverged apart due to the curvature of space."
"The Ricci curvature is the sum of all scalar curvatures in every basis vector direction."
"The volume of a ball isn't changing as it moves along geodesics if the Ricci curvature is zero."
"The Ricci curvature tracks the change in size of a ball as it travels along geodesics in space."
"You could certainly see the curvature of the Earth."
"The central idea behind a general theory of relativity is curvature."
"Let's just go here, one last thing we need to do is put the curved arc in there."
"Whenever we make a curve, we don't see the corner; it kind of disappears to the eye."
"What does it actually mean for space-time to be curved?"
"Our universe does not require an extra dimension to be curved into."
"Curvature tells us all the information we really care about when it comes to the geometry of a curve."
"Geodesic curvature is the amount by which the curve is going as straight along the surface as it possibly can while remaining on the surface."
"The principal curvatures describe the biggest and smallest rates of change or the greatest amount of bending in any direction."
"Surfaces with positive Gaussian curvature are convex surfaces; surfaces with zero Gaussian curvature are developable surfaces."
"The Gauss-Bonnet theorem says the total Gaussian curvature, if we integrate K over the whole surface M with respect to area, we get 2pi times something called the Euler characteristic Chi of the surface."
"Total curvature is a topological invariant of a surface."
"A geodesic is a curve where the geodesic curvature is zero."
"Curvature is a measure of how fast direction changes per unit of distance traveled."
"The curvature of a circle of radius a is one divided by the radius."
"If the curvature at some point p is equal to a, then the oscillating circle has a radius of 1 over a."
"Bending is the natural word for the second derivative on a graph."
"That means at that moment, it stopped bending down, and it's going to start bending up."
"It's called Gauss's Theorem Egregium, the remarkable theorem, and it relates Gaussian curvature to the curvature tensor."
"Shell structures are not just a form; they are very thin, can be membrane or folded, specially created with some curvature."
"The gradient tells you where to go; the Hessian tells you something about the curvature."
"Massive objects like stars cause deflections and curvature in space and time."
"We're going to use the fit point spline, and what this tool is going to do is it's going to allow you to create a really nice curvature."
"The most important thing is that we need a curve."
"The pressure always increases outward from the center of curvature."
"A curved two-dimensional surface is a reasonable prototype for Riemannian geometry in a positive sense."
"The bands of many of the sectional matrix kits out there today are curved in three dimensions to help recreate the anatomy of the tooth."