What is Curvature: Definition and 910 Discussions

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number.
For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.
For Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor.

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  1. S

    Question about spacetime curvature

    This maybe a simple question, but if Earth orbits the Sun due to the Sun's mass 'curving' spacetime, wouldn't we be moving closer to the sun? like if you spun a marble around within a bowl, it ends up in the center. What am I missing here?
  2. marcus

    Form of Friedman eqn with Λ curvature constant

    The basic equation of GR has a curvature constant Λ on the lefthand (geometric) side. The Friedman equation is derived from the Einstein Field Equation by making a simplifying assumption of uniformity. As a spacetime curvature Λ can be written either in units of reciprocal area or reciprocal...
  3. P

    MHB Can Gauss Curvature Be Zero on a Genus 2 Surface?

    Use the gauss bonnet theorem to show that the gauss curvature of a closed orientable surface of genus 2 cannot be identically zero euler characteristic is 2-2(2)=-2 so total gauss curvature is equal to -4pi. The integral of zero is zero and not -4pi so gauss curvature is not identically zero...
  4. marcus

    James Bond and the true value of the cosmological curvature constant

    Einstein discovered that general covariance allows his GR equation to have just TWO gravitational/geometric constants: Newton G and a curvature constant he called Lambda. So the symmetry of the theory requires us to put both constants into the equation and investigate empirically whether or not...
  5. P

    Finding Radius of Curvature of a Sphere Using Angle Excess

    On the surface of a sphere, we can find the radius of cuvature of the sphere by: angle excess / area = 1/ r_s^2 http://en.wikipedia.org/w/index.php?title=Angle_excess&oldid=543583039 If we use triangles, for instance, the angle excess is the sum of the angles of the triangle minus 180...
  6. N

    Confused about Vector Calculus Curvature Formulas? Let's Clear Things Up!

    Hey. so you have two formulas for curvature: The ordinary: |dT/ds| = |a|/|v|2 And the advanced: |v x a|/|v|3 = |a|*sin(α)/|v|2 = |aN|/|v|2 But the problem is, those two formulas aren't the same? The top one has acceleration divided by speed squared, while the bottom one has normal component...
  7. S

    Computing Curvature in 2D

    Homework Statement I have a given Metric: ds^{2}=A(u,v)^{2}du^{2}+B(u,v)^{2}dv^{2} And I'm asked to compute its curvature, and use this result to compute the curvature of the poincare metric: Set A=B=\frac{1}{v^{2}} The Attempt at a Solution I'm using Cartan's method. So first I change to an...
  8. A

    The Arc of Space Curvature: Large and Small

    When does the arc of the space curvature is large and when is it small?
  9. V

    Question about the curvature of spacetime:

    There's something very fundamental about the curved structure of spacetime that is confusing me. Einstein is saying that gravity can bend starlight. In other words, if I have this right, a star's light will follow the curvatures of spacetime created by a large body of mass, like the sun. Here's...
  10. M

    Second derivative of a metric and the Riemann curvature tensor

    I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial coordinates) R_{abcd}=\frac{1}{2} (g_{ad,bc}+g_{bc,ad}-g_{bd,ac}-g_{ac,bd}) where ",_i"...
  11. P

    Evidence of Electromagnetic fields causing spacetime curvature?

    Hi All, Just wanted to know, is there any experimental or observational evidence today, that electromagnetic fields can cause spacetime curvature? Either direct or indirect?
  12. R

    Principle curvature: min and max curvature are always perpendicular?

    I am trying to understand Gaussian curvature. This led me into looking at principle curvature. Now If one takes a look at the picture of the "Saddle Surface" on Wikipedia here: http://en.wikipedia.org/wiki/Principal_curvature I see that at the point p on the saddle where curvature goes both...
  13. V

    Thought experiment involving spacetime curvature:

    Peter Donis and Nugatory taught me a lot about spacetime curvature yesterday, but it has left me with so many questions. It sounds like mass slows down time as it warps spacetime. So, I suppose this means: more mass = more spacetime curvature = less time elapsing. Okay, in addition to...
  14. S

    Question: does the physical curvature of spacetime ever move ?

    Question: does the physical curvature of spacetime ever "move"? Something isn't adding up with Einstein's theory--or, more likely, I'm just not understanding it correctly! How can we say that the curvatures of spacetime created by the presence of stress-energy is giving us a continuum? When I...
  15. ssamsymn

    Calculating RiemannScalar in 2-D: Where to Start

    Where should I start from to show that curvature scalar (RiemannScalar) is 2\frac{R_1212}{det (g_μ√)} ?
  16. O

    Check if answers are right on my review? (vector eqns, curvature )

    Hi these are questions from my test review that i am unsure of, i posted question and my answer if you can tell me if I've gotten right answer that would be much appreciated! Let C be the curve with the equations x = 2 - t^3, y = 2t - 6, z = \ln(t) Find the point where C intersects the...
  17. E

    Space-time curvature as gravity only for speeding particles?

    If gravity rises from the fact that mass bends space-time and stuff falls in because it actually follows a straight line in a curved space as it moves by a gravitating object - doesn't that mean that a relatively stationary particle would not fall in the the claws of gravity as it would NOT be...
  18. A

    WHY does curvature of a convex lens affect focal length?

    Homework Statement For my high school physics coursework I must investigate factors affecting the focal length of a lens. I have focused on radii of curvature and completed my data collection and verified the accuracy using the lens makers equation. However, in the conclusion I am really...
  19. T

    Relative acceleration of geodesics and spacetime curvature

    Mass curves spacetime. The relative acceleration of nearby geodesics of free test particles indicates the sign of the spacetime curvature. Convergent geodesics mean positive, divergent negative curvature. But also the metric expansion of space curves spacetime. The geodesics may be convergent...
  20. B

    Finding The Curvature Of A Polar Function.

    Homework Statement Find the curvature of the polar function r = 5sin(2θ). Homework Equations All of the usual curvature equations. The Attempt at a Solution I want to turn this into a vector value function, so I can use the normal curvature equations, but that seems worse. I am...
  21. N

    How to calculate curvature of a vector in Mathematica.

    Homework Statement r(t)={(4+cos20t) cost,+(4+cos20t) sint,+0.4sin20t} Calculate the curvature of r[t] for 0≤t≤4pi Homework Equations k = | r' x r'' | / | r' |^3 The Attempt at a Solution r[t_]:={4+Cos[20t]*Cos[t],4+Cos[20t]*Sin[t],0.4Sin[20t]} k[t_]:=Norm[Cross[r',r'']]/Norm[r']^3...
  22. R

    No metric on S^2 having curvature bounded above or below by 0

    So I ran into a question; Show that there is no metric on S^2 having curvature bounded above by 0 and no metric on surface of genus g which is bounded below by 0. honestly I have no idea what is going on here. I know that a Genus is the number of holes in some manifold or the number of...
  23. 5

    Radius of a curvature for specific force on a seat in a jet

    The question is: What is the minimum radius of curvature of a jet, pulling out of a vertical dive at a speed of v, if the force on the pilot's seat is 7 times his weight? The way I thought to answer this is just to say that, 7 mg, the net force on the seat will be equal to the...
  24. Fernando Revilla

    MHB Mr.Ask's question at Yahoo Answers (curvature)

    Here is the question: Here is a link to the question: Find the curvature of the curve? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  25. X

    Gravity question: force vs spacetime curvature

    So Newton says that gravity is an attractive force and some people believe in gravitons to transmit that attractive force, but Einstein says the attraction is actually due to moving along the curvature of spacetime (caused by the bodies' mass). I'm not asking which is correct, but my question is...
  26. B

    A More Simple Curvature Proof.

    Homework Statement Prove the following statement K = (a(t) * N(t)) / (llv(t)ll)2 To clear things up it is the dot product of a(t), and N(t). Divided by the magnitude of velocity squared. Homework Equations llV X All / llV(t)ll3 The Attempt at a Solution I used the cross...
  27. B

    Proving Curvature Formulas: V X A / l V3l and a(t) * N(t) / l V(t) I2

    Homework Statement I have to prove two of the curvature formulas. The first one is (V X A) / l V3l The other one is a(t) * N(t) / l V(t) I2 Homework Equations I have a hint from my professor, but it is all confusing. I need a youtube video or something to get started on these...
  28. P

    Dark energy and dark matter as curvature effects?

    The so called f (R)-gravity could be, in principle, able to explain the accelerated expansion of the Universe without adding unknown forms of dark energy/dark matter but, more simply, extending the General Relativity by generic functions of the Ricci scalar. However, a part several...
  29. R

    Gravitation: Curvature- vs -particles

    On occasion I notice there is some talk about "graviton" particles, I would have thought astro/ quantum sciences were past that idea. I am quite aware of a basic rule "Don't fall in love with your theories" so a gravity particle might exist, more on that later. In my understanding of...
  30. R

    What is the relationship between the limit and curvature of a trajectory?

    Hi! Can someone please explain how does the limit in the attachment equals the curvature of a trajectory? I do not understand it. Why is it defined this way? ζ=dx/dl and it is in the direction of T. Thank you!
  31. J

    Curvature of Spacetime on Earth

    I am trying to improve my understand of the basic elements of GR. I have read that the Earth orbits the sun because spacetime between the Earth and the sun is warped, mainly due to the sun’s mass. The Earth follows a geodesic, which is the equivalent of a straight line in curved space...
  32. micromass

    Geometry Riemannian Manifolds: An Introduction to Curvature by Lee

    Author: John Lee Title: Riemannian Manifolds: An Introduction to Curvature Amazon link https://www.amazon.com/dp/0387983228/?tag=pfamazon01-20 Prerequisities: "Introduction to Smooth Manifolds" by Lee seems like a prereq. Level: Grad Table of Contents: Preface What Is Curvature? The...
  33. anorlunda

    Both positive and negative curvature?

    What happens to the Reimann tensor at the event horizon of a black hole? Do some of the 24 components become zero or infinite? What happens to parallel transport of a vector on the surface of an event horizon that is different than on a surface outside the event horizon? I'm newly educated...
  34. M

    Computing Curvature in 3 space

    Homework Statement Consider the hyperboloid x2+y2-z2 = 1 at the point (1,0,0). Take the normal direction i to the surface. a) Compute the curvature of the circle x2+y2=1 on the hyperboloid (z=0) at the point (1,0). b) Compute the curvature of the hyperbola x2-z2=1 on the hyperboloid...
  35. L

    Gaussian Radius of curvature in an argon ion laser

    Hi, having a bit of trouble with this question "In an argon ion laser ( λ = 514nm) the minimum beam waist is 1.0mm and is close to the plane mirror. Calculate the radius of curvature of the beam at the output mirror. 1.15m away" Attempt at a solution: θ = 2λ/pi W02 R = Z + ZR2/Z ZR =...
  36. shounakbhatta

    Mass causes space time curvature

    If I take it by literally meaning: Mass causes space time to curve. A rubber sheet where the mass is there, it causes the dent, the curvature. So it means the greater the momentum, the greater the curve or the dent. Now if we have a very big mass, I mean to say big in terms of size, the...
  37. B

    What went wrong in calculating the curvature for r(t)=<t^2,lnt,tlnt>?

    Homework Statement r(t)=<t^2,lnt,tlnt> Homework Equations k= |T '(t)| / |r '(t)| The Attempt at a Solution My professor's answer sheet solved the problem using the other method, k(t)=|r '(t) x r ''(t)| / |r '(t)|^3 and that answer ends up being 0.3, while mine is 0.4. I...
  38. G

    What is the measure of space-time curvature and how is it calculated?

    Space-Time Curvature Question! Hi Guys, A question about the curvture of space-time by mass. Where is the point of maximum curvature?? Is it at the centre of mass (i.e.. the middle of the body) The reason I ask, is that when space-time curvature is shown visually it makes out like it is a...
  39. N

    (wald) method for calculating curvature

    R_{a}_{b}_{c}^{d}ω_{d}=((-2)\partial_{[a}\Gamma^{d}_{b] }_{c}+2\Gamma^{e}_{[a]}_{c}\Gamma^{d}_{[b]}_{e})ω_{d} good, me question is about of: 1.- as appear the coefficient (-2) und the (2)? 2.- it is assumed that...
  40. B

    Moments of deflection/ curvature of radius of a supported beam

    I was wondering if there were any mechanical engineers that can answer a few questions I have regarding an assignment that I have been set. We have to choose a suitable beam to support a monorail. we are looking for a moment of deflection of around 10mm. Using the universal beams table bs 4 1993...
  41. D

    Why positive curvature implies finite universe?

    This post in influenced by 3 new threads in our cosmology forum. Recent observational data favors positive curvature of our Universe. The question I have, however, is why positive curvature implies spatially finite Universe? Yes, it might look quite obvious if we embed curved space into higher...
  42. N

    Integral of mean curvature function

    Hello everyone, I am self teaching some elementary notions of differential geometry. Rather, I should say I am concentrating on mean and gaussian curvature concepts related to a physics application I am interested in. I see one has to evaluate an integral that goes as...
  43. A

    Spaces of Constant Curvature and the Ricci Tensor

    Hi all, I was just interested in verification of a concept. If we are given the full Riemann tensor in the form which implies constant curvature (i.e. lambda multiplying metric components) does this imply that the Ricci tensor vanishes? The question stems from why the vacuum equations cannot be...
  44. G

    Double contraction of curvature tensor -> Ricci scalar times metric

    Double contraction of curvature tensor --> Ricci scalar times metric I'm trying to follow the derivation of the Einstein tensor through double contraction of the covariant derivative of the Bianchi identity. (Carroll presentation.) Only one step in this derivation still puzzles me. What I...
  45. F

    Understanding the Effects of Relative Curvature and Mass on Space and Observers

    Mass curves space. And speed near the the speed of light increases mass. So for someone traveling near c and is passing a partice at rest, the traveling observe feels like he's at rest and the other particle is moving. So if the other particle is moving wrt his rest frame, does he see an...
  46. F

    Conditions for curvature singularity

    Hi All, I was wondering if it is correct to say that a vanishing metric determinant is a necessary (but probably not sufficient) condition for a curvature singularity to exist at some point(s), or is one forced to construct the full Kretschmann scalar? Cheers! FD
  47. C

    Curvature of an orthogonal projection

    Homework Statement Let \vec{X(t)}: I \rightarrow ℝ3 be a parametrized curve, and let I \ni t be a fixed point where k(t) \neq 0. Define π: ℝ3 \rightarrow ℝ3 as the orthogonal projection of ℝ3 onto the osculating plane to \vec{X(t)} at t. Define γ=π\circ\vec{X(t)} as the orthogonal projection...
  48. G

    Radius of Curvature calculation in a magnetic field

    Homework Statement 1) The magnetic field everywhere is tangential to the magnetic field lines, \vec{B}=B\hat{e}t, where \hat{e}t is the tangential unit vector. We know \frac{d\hat{e}t}{ds}=(1/ρ)\hat{e}n , where ρ is the radius of curvature, s is the distance measured along a field line and...
  49. P

    Radius of Curvature, Lenses, Finding Di

    Homework Statement If you look at yourself in a shiny Christmas tree ball with a diameter of 8.1 cm when your face is 35. 0cm away from it, where is your image? Homework Equations 1/do + 1/di = 1/f The Attempt at a Solution 1/di = 1/4.05 cm - 1/35.0 cm di = 4.6 cm I...
  50. A

    How to calculate radius of curvature of beam?

    If I start out with a flat beam of length a and then I fix one side and then bend the other side up to form an arc with height h, is that enough information to determine the radius of curvature of the bent beam? If so, how would I do it? Thanks...
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