What is Curvature: Definition and 912 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. K

    Space-time curvature without mass-energy ?

    Hi everybody, As you know, the Einstein field equation R_{μ\nu} - 1/2Rg_{μ\nu} =κT_{μ\nu} implies that at any point with vanishing energy-momentum tensor the Ricci curvature also vanishes: T_{μ\nu} = 0 \Rightarrow R_{μ\nu} = 0 hence a Ricci-flat space-time (the vacuum...
  2. S

    Understanding the Ricci Curvature Tensor in Einstein's Field Equations

    I've been studying the Einstein field equations. I learned that the Ricci curvature tensor was expressed as the following commutator: [∇\nu , ∇\mu] I know that these covariant derivatives are being applied to some vector(s). What I don't know however, is whether or not both covariant...
  3. E

    Space/time curvature of the young universe?

    All over the news we see the results of the recent detection of gravity waves from the early universe. Which got me wondering: The early universe was much more dense than at the present. It therefore seems that spacetime was much more curved than it is, on average, today. Is this...
  4. marcus

    Length and curvature operators in Loop gravity

    Most of us are familiar with the fact that in Loop gravity the area and volume observables have discrete spectrum. The discrete spectrum of the area operator, leading to a smallest positive measurable area, has lots of mathematical consequences that have been derived in the theory. It helps...
  5. C

    Curvature of flexible chain

    Dear all, I was wondering how the radii of curvature can be calculated of a flexible chain (polymer chain). I have the x,y and z values of the polymer chain. For a 2D chain, I can calculate the curvature radii (http://www.intmath.com/applications-differentiation/8-radius-curvature.php). I am...
  6. M

    Parametric equations for circle of curvature at given point.

    Hey guys, I'm new here. I got a problem from my professor that is different from any other problems we have done. I'm stuck and need a little help.Homework Statement r(t) = <cos(t), t, 2sin(t)> Find parametric equations for the circle of curvature at (0, pi/2, 2)The Attempt at a Solution I...
  7. Demon117

    Geodesic curvature, normal curvature, and geodesic torsion

    I am struggling to make sense out some things. Hopefully someone can help or at least offer some different point of view. Let's examine a differential curve parameterized by arc length that maps some interval into an oriented surface (lets call it N(s)). The surface has a unit normal field...
  8. shounakbhatta

    Gravity as a force and as a curvature

    Hello All, As far as the Newtonian mechanics and Einstein's GR is concerned, I am a little bit confused in the following things: (a) Concerning the bending of light due to gravity: Some lectures and opinions show that light bends due to the force of gravity as shown in the event of a solar...
  9. S

    Science fictiony questions about spacetime curvature

    I'm writing a sci-fi story and I'd like to make it, at the very least, scientifically plausible (in the way that alcubirre warp drives are possible assuming we could get our hands on something with negative mass which, as far as we know, doesn't exist). The basic assumption for these questions...
  10. Markus Hanke

    Is Weyl Curvature Present in Interior Spacetimes?

    I am just wondering - is space-time curvature in the presence of energy-momentum ( i.e. in interior solutions to the EFEs ) always pure Ricci in nature ? I had a discussion recently with someone who claimed that, but personally I would suspect that not to be the case in general, since I see no...
  11. L

    What is the Radius of Curvature at Point B on the Road?

    Homework Statement The speed of a car increases uniformly with time from 50km/hr at A to 100km/hr at B during 10 seconds. The radius of curvature of the bump at A is 40m. if the magnitude of the total acceleration of the car’s mass center is the same at B as at A, compute the radius of...
  12. R

    Curvature of Space: What is it?

    Curvature of Space-time: What is it? General relativity talks about curvature of space-time due to mass. What does it actually mean by 'curve'. Is the space made of something that we can say is curving? If space is purely 'empty', then what is getting curve? Or is it that curving is just an...
  13. P

    Equation for parallel transport involving sectional curvature

    This is an expression I came across in a paper I am going through. It involves an expression for the parallel transport of a tangent vector taking into consideration the sectional curvature of simply connected space-forms in \mathbb R^4 . I have not been able to derive it.The equation simply...
  14. B

    Gravity and the curvature of space-time

    This is something I have pondered for some while... it is so obvious that there must be an answer and is probably a silly question, but I haven't found an answer yet... so... Gravity is a consequence of the localised curvature of space. According to Relativity, space (space-time) is...
  15. Markus Hanke

    Kretschmann Scalar: Physical Significance & Invariants of Riemann Curvature

    I have a basic question regarding the invariants that can be formed from the Riemann curvature tensor, specifically the Kretschmann scalar. Does this quantity have any physical significance, in the sense that it is connected to anything physically measurable or observable ? My current...
  16. S

    Calculate the curvature and vectors T,N,B

    Homework Statement Let ##\Gamma ## be trajectory which we got from ##z=xy## and ##x^2+y^2=4##. Calculate the curvature ##\kappa ## and vectors T, N and B (B is perpendicular to T and N). Homework Equations The Attempt at a Solution Well, the hardest part here is of curse to...
  17. P

    Single equation for space-time curvature?

    Is there a single equation that can model both spatial and temporal metric contraction simultaneously? And also what's that equation that can model the actual degree of curvature n space-time that uses trig functions and how do you use that in combination with the two previously mentioned...
  18. T

    Pre-inflationary spatial curvature

    Prior to expansion the inflaton field had a large potential energy. I wonder whether there are any considerations or calculations to evaluate how to this energy curves the space created by the big bang. Does it make sense at all to talk about critical vs. actual energy density, the value of...
  19. S

    Does a compact manifold always have bounded sectional curvature?

    Sorry if this question seems too trivial for this forum. A grad student at my university told me that a compact Riemannian manifold always has lower and upper curvature bounds. Is this really true? The problem seems to be that I don't fully understand the curvature tensor's continuity etc...
  20. B

    Why the curvature of spacetime is related to momentum?

    Well, I'm totally in a mess now
  21. J

    Does the curvature of spacetime have a unit?

    Does the curvature of spacetime have a unit?
  22. X

    Homework help - universe density & curvature

    1. If the current mass density in the Universe was about 10 protons/m3 what would be the current radius of its curvature? What would be the maximum distance between the two points in the Universe? I got the first part but not the 2nd. If I solve the Friedman equation I get the max scale factor...
  23. Q

    Curvature of space and spacetime

    i am trying to understand the relationship between the two on a local and global scale and how these two concepts are related to the Ricci scalar. Is it correct to say that as far as we know on a global scale, spacetime is flat so that the Ricci scalar is zero. If so, what can be said about...
  24. N

    Radius of curvature of steel rod under stress

    Hi, I have a problem which you guys probably could help me solve or at least advise how to approach. I am building a mechanical system that consists of 2 steel rods acting as rails and a platform that travels along. I need to find radius of curvature of a steel rod under stress to see by...
  25. K

    Is it possible for a wormhole to have zero spacetime curvature?

    I have always read that a wormhole will quickly collapse in on itself due to its own gravity, forming a black hole, unless it is held open by some exotic matter that has a negative energy density. But couldn't there exist a wormhole with zero spacetime curvature? It would therefore have no...
  26. I

    Curvature vs acceleration? (calc III)

    i asked this question before, but i didn't ask it quite right so i didn't get a satisfactory answer.. curvature is define as how quickly/ abruptly a curve changes with respect to its arc length.okay so the normal vecor (N = T ') is the change in the tangent vector of a curve with respect to...
  27. L

    Finding Curve w/ Curvature 2, Passing Through (1,0)

    Homework Statement Find the curve whose curvature is 2, passes through the point (1,0) and whose tangent vector at (1,0) is [1/2 , (√3)/2 ]. The Attempt at a Solution I know I must use the Fundamental theorem of plane curves but I don't know how to apply it correctly here.
  28. H

    Does Mass or Size Affect Curvation in Relativity?

    General Relativity says that every object that has mass make a curvation to space-time. Ι want to ask from what depends the curvation. Only from his mass? It depends from the size of object? For example let's say that we have one object with 10 meter size,and an other with 1000,but they have...
  29. D

    MHB Curvature and torsion on a helix

    Consider the case of a right circular helical curve with parameterization \(x(t) = R\cos(\omega t)\), \(y(t) = R\sin(\omega t)\), and \(z(t) = v_0t\). Find the curvature and torsion curve. http://img30.imageshack.us/img30/7828/gwi.png We can then parameterize the helix \begin{align*}...
  30. J

    What is the radius of curvature formula for an ellipse at slope = 1?

    What is the radius of curvature formula for an ellipse at slope = 1? I have found b^2/a, and a^2/b for the major and minor axis, but nothing for slope = 1. Thanks.
  31. S

    Compact 3-manifolds of Negative Curvature

    Does anyone know if it is possible to construct a compact 3-manifold with no boundary and negative curvature? I ask this question in the Cosmology sub-forum because I see in various writings of cosmologists that it is often taken for granted that a negatively curved Universe must be infinite...
  32. I

    Curvature of Space: What is it?

    What is
  33. S

    Curvature Tensor: Non-Zero in Local Inertial Frame

    hi In a local inertial frame with g_{ij}=\eta_{ij} and \Gamma^i_{jk}=0. why in such a frame, curvature tensor isn't zero? curvature tensor is made of metric,first and second derivative of metric.
  34. E

    Solving Scalar Curvature for Homogenous & Isotropic FLRV Metric

    Homework Statement Find the equation of scalar curvature for homogenous and isotropic space with FLRV metric. Homework Equations ## R=6(\frac{\ddot{a}}{a}+\left( \frac{\dot{a}}{a}\right )^2+\frac{k}{a^2}) ## The Attempt at a Solution ##G_{AB}=R_{AB}-\frac{1}{2}Rg_{AB}##
  35. E

    What is the Riemann Curvature Tensor for Flat and Minkowski Space?

    Homework Statement Show that all components of Riemann curvarue tensor are equal to zero for flat and Minkowski space. Homework Equations The Attempt at a Solution ## (ds)^2=(dx^1)^2+(dx^2)^2+...+(dx^n)^2 \\ R_{MNB}^A=\partial _{N}\Gamma ^{A}_{MB}-\partial _{M}\Gamma ^{A}_{NB}+\Gamma...
  36. I

    What is curvature? (multivariable calculus)

    if space curve C=<f(t),g(t),h(t)>, and v=\frac{dC}{dt}=<\frac{df(t)}{dt},\frac{dg(t)}{dt},\frac{dh(t)}{dt}> Why is curvature defined this way? κ\equiv\frac{d\widehat{T}}{dS} \hat{T}=unit tangent vector S=arc length to elaborate, for a space curve, i understand what \frac{dT}{dt} is, but what...
  37. S

    Smallest curvature radius in gradient index optics

    Hello, would someone know what is the smallest radius of curvature achievable with current gradient index optics (GRIN) technology? I mean, how much could one "curve" a ray of light? Many thanks! :smile:
  38. R

    Gravitation as curvature of space vs field theory

    Gravitation is described on one hand as curvature of space in the presence of matter. It is also described as a field acting through gravitons on matter. How can the two views be reconciled?
  39. B

    Finding The Coordinates of The Center Of Curvature

    Homework Statement Let C be a curve given by y = f(x). Let K be the curvature (K \ne 0) and let z = \frac{1+ f'(x_0)^2}{f''(x_0)}. Show that the coordinates ( \alpha , \beta ) of the center of curvature at P are ( \alpha , \beta ) = (x_0 -f'(x_0)z , y_0 + z) Homework Equations The...
  40. S

    Why do objects fall through curvature of spacetime?

    Just wondering, if the way to describe the movement of objects through spacetime is to say that they fall through the curves created in 4D spacetime, then is it a stupid question to ask why objects don't rise through spacetime? Or is this the same thing and rising and falling are one of the same...
  41. Z

    Riemann curvature scalar, Ricci Scalar.What does they measure ?

    hello Can you perhaps explain what does the Riemann curvature scalar R measure? or is just an abstract entity ? What does the Ricci tensor measure ? I just want to grasp this and understand what they do. cheers, typo: What DO they measure in the title.
  42. B

    Finding Curvature of Vector Function

    Homework Statement Find the curvature K of the curve, where s is the arc length parameter: \vec{r}(t) = \langle 2 \cos t , 2 \sin t, t \rangle Homework Equations s(t) = \int_a ^t ||\vec{r}'(u)||du The Attempt at a Solution I know I need to find the arc length function, in order to find the...
  43. Z

    Riemann Curvature Scalar Differs in Landau & MTW

    hello For the same Friedmann metric, Landau (Classical theory of fields) finds a value for the Riemann curvature scalar which is given in section 107 : R = 6/a3( a + d2(a)/dt2) whereas in MTW , in box 14.5 , equation 6 , its value is : R = 6(a-1 d2(a)/dt2 + a-2 (1 + (d(a)/dt)2 ) ) The...
  44. A

    Second fundamental form and Mean Curvature

    Homework Statement Metric ansatz: ds^{2} = e^{\tilde{A}(\tilde{\tau})} d\tilde{t} - d\tilde{r} - e^{\tilde{C}(\tilde{\tau})} dΩ where: d\tilde{r} = e^{\frac{B}{2}} dr Homework Equations How to calculate second fundamental form and mean curvature from this metric? The Attempt at a...
  45. K

    Understanding Positive and Negative Intrinsic Curvature in General Relativity

    if gravity arises from normal accelerations due to the curvature of spacetime...what would the opposite of this "process" represent? to clarify is it possible to describe the opposite of this curvature?? thanks
  46. I

    Simplifying the Algebra for Gaussian Curvature

    Homework Statement Hello everyone. :) I'm having trouble simplifying the last little bit of this question that deals with Gaussian curvature. I've taken all the required derivatives, and double checked with my professor to make sure that they're correct. I'm only have trouble with reducing...
  47. S

    Type of curvature of gradient force from edge to center of a sphere

    I was doing some simple physics with a ball resting on a table and I made this curve (0,0) (25, 6.8) (50, 27.51) (75, 63.4) (100, 112.34) (125, 175.7) (150, 253.3) (175, 345.4) I was wondering if anyone could identify what sort of curve it is? I am just curious. This is not a homework...
  48. T

    Torsion and Curvature: Understaind The Theory of Curves

    I don't really understand the point in Curvature and Torsion, I am wondering if someone could explain them to me. Thank you for your kindness: Why do mathematicians need Curvature and Torsion? What are their main uses??
  49. T

    Is gravity a force, or the curvature of space?

    After some light reading, I'm more confused than ever. Is gravity just a byproduct or effect of the curvature of space? Is it a force that would exist if space didn't curve, even in the presence of mass? (probably a stupid question, sorry!) I've seen various diagrams of the Earth revolving...
  50. 4

    Is there always the same amount of spacetime curvature in the uni.?

    Is there always the same "amount" of spacetime curvature in the uni.? Universe is what I meant by uni. Okay, if matter and energy cannot be created or destroyed, and since they are what causes spacetime to curve, does that mean there will always be the same amount of spacetime curvature...
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