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. Pouramat

    Exercise 16, chapter 3 (Tetrad) in Carroll

    My attempt at solution: in tetrad formalism: $$ds^2=e^1e^1+e^2e^2+e^3e^3≡e^ae^a$$ so we can read vielbeins as following: $$ \begin{align} e^1 &=d \psi;\\ e^2 &= \sin \psi \, d\theta;\\ e^3 &= \sin⁡ \psi \,\sin⁡ \theta \, d\phi \end{align} $$ componets of spin connection could be written by using...
  2. italicus

    I Radius Excess of Curvature in Relativity: Explained

    I think the best place to put this post is the section on special and general relativity. Reading Feynman’s lecture n.42 , volume II here linked : https://www.feynmanlectures.caltech.edu/II_42.html I’ve met the following formula 42.3 for the radius excess of curvature, that Feynman attributes...
  3. E

    I Riemann Curvature Tensor on 2D Sphere: Surprising Results

    I have worked out (and then verified against some sources) that ##R^\theta_{\phi\theta\phi} = sin^2(\theta)##. The rest of the components are either zero or the same as ##R^\theta_{\phi\theta\phi} ## some with the sign flipped. I was surprised at this, because it implies that the curvature...
  4. George Keeling

    I Const Curvature Scalar & 3-Torus: Is It Maximally Symmetric?

    Spatial slices of the Robertson-Walker metrics are maximally symmetric so they must have a constant curvature. Is it true that in three Riemannian dimensions that a constant curvature scalar determines whether the volume is finite or infinite? Carroll seems to have given a counter-example for...
  5. Martian2020

    B General Relativity and the curvature of space: more space or less than flat?

    General relativity. Curvature of spacetime: ok. time dilation: ok. What about space? Curvature is intrinsic and given by complex equations. But could we definitely say is there more space between 2 points along curved space through the star than would be through flat space (no star there) or...
  6. Martian2020

    B Visualize 2D Intrinsic Curvature of Spacetime (1s+1t) in 3D

    Via web search found https://www.physicsforums.com/threads/what-dimension-does-space-time-curve-in.852103/ Read it and watched two videos mentioned: I understand we cannot perceive 5D ;-), so extrinsic visualization of maximum of 2D intrinsic curvature is possible. So time+1d space is all we...
  7. S

    A What is the role of Berry curvature in solids according to Rev. Mod. Phys?

    This is a paragraph from Rev. Mod. Phys,82,1959(2010). From the article, I understand that Berry phase is gauge invariant only for closed loop evolution but what exactly is this evolution? Does it mean that the system initially start out with some Hamiltonian and I continuously change the...
  8. T

    B Can Curvature and Geodesics be Generalized in Higher Dimensions?

    Hello there.Curvature can be informally defined as the deviation from a straight line in the context of curves, a circle in R^2 has curvature, then if we get higher dimensions than three we can't see the manifolds because it is their nature and the nature of our eyes that it is bounded by the...
  9. faca

    B Curvature radius question for my muon beam experiment

    Hello, I've a particle beam moving along the z-axis. The beam goes through a dipole magnet. I studied the hit position in a tracker after the magnet and I noticed that there are hits at 2 different x coordinate (the x asix is transverse to the z one). Let's call Delta x the shift between the 2...
  10. Buzz Bloom

    I Question re spacial curvature K(r) w/r/t the Shwarzchild metric

    I understand that K(∞) = 0, and K(rs) = ∞ where rs = 2GM/c2. What is an equation for K(r) when rs < r < ∞? I have tried the best I can to search the Internet to find the answer, but I came up empty. I would very much appreciate the answer, or a reference that discusses the desired answer. I...
  11. P

    A Curvature Tensor for Dual Vectors

    Good day all. Given that in Sean Carroll`s Lectures on GR he states that when calculating the covariant derivative of a 1-Form the Christoffel symbols have a negative sign as opposed to for the covariant derivative of a vector, would it be naive to think that, given the usual equation for the...
  12. Arman777

    I Curvature of Space in the Context of Cosmology

    Recently I asked a question about the curvature of the universe. https://www.physicsforums.com/threads/constant-curvature-and-about-its-meaning.977841/ In that context I want to ask something else. Is this curvature (##\kappa##) different than the Gaussian Curvature ? Like it seems that we...
  13. L

    I Derived metrics on surfaces of positive curvature

    Start with a closed surface of positive Gauss curvature embedded smoothly in ##R^3##. At each point, choose two independent eigenvectors of the shape operator whose lengths are the corresponding principal curvatures. By declaring them to be orthonormal one gets - I think - a new metric on the...
  14. Z

    Can You Detect the Earth's Curvature from a Lighthouse?

    If you stood on a lighthouse high above with a very accurate transit, would you be able to detect the curvature of the earth?
  15. DarkMattrHole

    I Accelerating Body Creates Space-Time Curvature?

    If you are floating in space in your spaceship and you kick in the engines and accelerate at a comfortable 1G and you end up standing on the bottom of your ship, a slight curvature of space-time is formed, throughout the ship, perhaps immeasurable, such that without windows on the ship, you...
  16. J

    I Is Zero Curvature Space Equivalent to Flat Space in General Relativity?

    In relativity, a flat space is always regraded to be endowed with an invariant metric field ##g_{\mu\nu}(x)= \eta_{\mu\nu}##, So in a flat space the corresponding connection ##\Gamma_{\mu\nu}^\rho(x)=0## It means that if we parallel transport a vector ##v^\nu(x_0)## in the space. Then it...
  17. P

    A Parallel transport of a 1-form aound a closed loop

    Good day all. Since the gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Then If we form the Gradient vector field...
  18. Y

    Engineering Finding the radii of curvature for an achromatic lens

    Since the two lenses are plano-concave and plano-convex, the lens maker equation can be simplified into containing only 1 R for each of the lens. Substituting the values for the red light, I got: 1/f1=0.51/R1 and 1/f2 = 0.64/R2. Adding these two equations and equating them to 1/(500*10-3)...
  19. Rabindranath

    A Weyl transformation of connection and curvature tensors

    Given a Weyl transformation of the metric ##g_{\mu\nu} \rightarrow g'_{\mu\nu} = e^{\Omega(x)} g_{\mu\nu}##, I'm trying to find the corresponding connection ##\Gamma'^{\lambda}_{\mu\nu}##, and from that ##-## via the Riemann tensor ##R'^{\lambda}_{\mu\nu\kappa}## ##-## the Ricci tensor...
  20. G

    I General Relativity & Curvature Explained: How Objects Fall

    I always was confused on how objects fall on Earth due to curvature of space. All I see everywhere is a flat stretched piece of fabric and balls going round and round the central massive ball. But that does not explain anything, how objects actually fall. Finally I saw this video where it...
  21. Castty

    B Infinite Curvature & Hawking Radiation Explained

    Hi, i have a question which i can't solve myself, as i am not a student of physics: I have heard of the infinite space curvature which occurs when matter collapses into a black hole. On the other hand i have heard, that a black hole radiates energy away. Now i see a contradiction: When the...
  22. Arman777

    I Clarifying the Meaning of Radius of Curvature in Cosmology

    $$1 - \Omega_{tot} = \Omega_κ = \frac{-κc^2}{R_0^2H_0^2} $$ For ##\Omega_κ=-0.0438## we get a some value for ##R_0##. This ##R_0## is the radius of the observable universe right ? Not the universe ?
  23. E

    How can I use the covariant derivative to derive the Riemann curvature tensor?

    I derived this equation $$ A_{i,jk}-A_{i,kj}=R^r _{kij}A_r$$.But where do I use this $$A_{i,j}+A_{j,i}=0$$?
  24. C0nstantine

    A Diffeomorphic manifolds of equal constant curvature

    Every two semi-Riemannian manifolds of the same dimension, index and constant curvature are locally isometric. If they are also diffeomorphic, are they also isometric?
  25. Arman777

    I Constant Curvature and about its meaning

    In a book that I am reading it stated that, the constant curvature implies curvature is homogeneous and isotropic, hence only three ##κ## values are possible for our universe $$κ = -1, 0, +1$$ as we all know these values represent negative, flat and positive curvature respectively. Now if...
  26. Nimarjeet Bajwa

    Why can't we do this by using radius of curvature?

    I have tried this question thrice. and for 3 days. I will try to explain My attempts as best as i can Attempt-1--> This is fairly basic. I found X(t) and Y(t) in polar form and put them in the equation of circle. After that diffrentiated both sides with respect to "x" however the answer came...
  27. M

    Helicoidal movement: acceleration vector, arc length, radius of curvature

    I have tried to solve it and I would like a confirmation, correction or if something else is suggested... :) Helicoidal movement
  28. Like Tony Stark

    Calculating the radius of curvature given acceleration and velocity

    Well, what I've done so far is calculating the magnitude of velocity and acceleration replacing ##t=2## in ##\theta (t)## and ##r(t)## so I could get the expressions for ##\dot r##, ##\dot \theta##, ##\ddot r## and ##\ddot \theta##. But that's not my problem... my problem is related to the...
  29. MatthijsRog

    I Visualizing Intrinsic Curvature: Is This a Good Practise?

    I’m working through the books by Schutz and Renteln to get my differential geometry to the point where I can do general relativity. The author’s have just introduced metrics and notions of parallel transport and with some work I finally understand how intrinsic curvature can be defined by...
  30. W

    I Does Spatial Curvature Falsify Eternal Inflation?

    I have often heard it said that the picture of a multiverse inspired from eternal inflation is not falsifiable. However this paper from 2012 claims that it is: https://arxiv.org/pdf/1202.5037.pdf specifically it says, as I understand it, that if one were to measure sufficient positive spatial...
  31. Ranku

    I Spacetime curvature and curvature index

    The presence of the cosmological constant produces a flat spacetime universe with Ω = 1. There is also the curvature index of space k, which can be +1, 0, -1. But it is possible to have any of these values of k with Λ > 0 or Λ < 0. How is the curvature of spacetime determined by Λ different from...
  32. QuarkDecay

    A Different FRW Cosmological Models

    ρο/a4 ρο/a3 k Λ Βig Bang Big Crunch Polynomial Expansion Exponential Expansion x 0 x 0 >0 x +1 x +1 x 0 <0 x -1 x -1 >0...
  33. W

    Riemann Curvature Tensor in 2D

    Since in 2D the riemman curvature tensor has only one independent component, ## R = R_{ab} g^{ab} ## can be reversed to get the riemmann curvature tensor. Write ## R_{ab} = R g_{ab} ## Now ## R g_{ab} = R_{acbd} g^{cd}## Rewrite this as ## R_{acbd} = Rg_{ab} g_{cd} ## My issue is I'm not...
  34. H

    How do I find the radius of curvature?

    I figured out that the focal distance is 19.01868
  35. Papo1111

    B Traveling using space curvature

    Hi everyone. I'd like to verify my thoughts about travellig through space using a space curvature. Imagine you have a spaceship and you want to travel some distance. Your ship launches an object into space that has huge mass and density. It curves space. Now, you enter the curved space and...
  36. redtree

    I Fourier transform on manifolds

    Is there a generalized form of the Fourier transform applicable to all manifolds, such that the Fourier transform in Euclidean space is a special case?
  37. H

    A Variational derivative of mean curvature

    Hello Physics Forums, I have a simple parametric surface in R3 <x,y,z(x,y)>. I've calculated the the usual mean curvature: H= ((1+hx^2)hyy-2hxhyhxy+(1+hy^2)hxx)/(1+hx^2+hy^2)^3/2 I needed to take the variational derivative of this expression. Since it has second order spatial derivatives the...
  38. N

    I Space-time curvature and the fabric of space

    Greetings: I watched several videos describing so-called "empty space" as being permeated with fields (electron field, quark field, etc.). Is it possible that it is actually these fields that curve about large masses and that the trajectory of light and matter curve because they follow the...
  39. Osvaldo

    I Spacetime Curvature: Eliptical Orbital Paths & Keppler Laws

    Though it is hard not to believe in the spacetime curvature that cause planets to follow curved path arround massive objects, I wander how come these paths are eliptical, the object change velocity when moving arround the massive object and what is more obeys the Keppler laws. If there is not...
  40. D

    Radius of curvature of the trajectory of points A and B

    Homework Statement A cylinder rolls without slippage on a horizontal plane. The radius of the cylinder is equal to r. Find the radious of curvature of the trajectory of points A and B. Homework Equations Ciruclar motion equations. ##R=\frac{1}{C}## The Attempt at a Solution First I drew the...
  41. Buckethead

    B Spacetime curvature due to acceleration causing gravity?

    In reviewing one of Einstein's thought experiments, the accelerating elevator in space, and the resulting bending of light passing through the elevator, Einstein's predicted that light will bend in gravity. Now Einstein's original prediction was off by a factor of 2 because he hadn't yet...
  42. R

    Why Centrifugal Force in the Derivation of Curvature Drift

    Homework Statement This is a question about a pretty basic plasma physics derivation. In the standard derivation of the Curvature Drift of a charged particle in a magnetic field with curvature, the force that they use is the "imaginary" centrifugal force (or the force the guiding center sees in...
  43. Arman777

    I Relationship between Energy density and Curvature

    I don't know GR so while answering the question if you prefer not to use that, I would be happy. In the Friedmann Equations, is energy density has an effect on curvature or vice versa? Or they are separate things and they don't affect each other? For example can we have an energy density...
  44. Arman777

    I Why does an Empty Universe have to obey Negative Curvature?

    Its stated that empty universe should have a hyperbolic geometry (Milne Universe) but I don't understand how its possible. $$H^2=\frac {8\pi G\epsilon} {3c^2}-\frac {\kappa c^2} {R^2a^2(t)}$$For an empty universe when we set ##\epsilon=0## we get $$H^2=\frac {-\kappa c^2} {a^2(t)}$$...
  45. H

    I Curvature of space in large regions: zero or not?

    I have read numerous times that the overall curvature of space in extremely large regions -1000s of megaparsecs say - is zero. I also keep reading that the expansion rate of the universe is increasing, and that the universe is resultantly positively curved. I would be interested in a...
  46. A

    B Question regarding if curvature is not constant

    For 2-sphere it is having a curvature of k=1/R ,where R is the radius of the 2-sphere and to make it more generalised we treat the kR as the curvature which is always +1 and is independent of it's radius. My question is how do we treat the curvature term for 3-sphere , And it the curvature term...
  47. A

    B Positive curvature of a 2D surface

    If considering a 2D surface (plane) having polar coordinate r,θ (where r is the distance from the origin and θ is the anticlockwise angle from the base line as usual) The metric is now actually ds2=dr2+r2dθ2 If now this 2D surface is given a positive curvature of +1 (equivalent to the surface of...
  48. Hugh de Launay

    I Does S-T Curvature Change Particle Shape?

    This is an inquiry about some of the details of the bending of the trajectory if a photon by curved space-time (S-T) from the perspective of general relativity (GR). The environment in this scenario contains a black hole at its center, a photon passing by the black hole a km above the event...
  49. P

    A Fundamental definition of extrinsic curvature

    My question is quite simple: what is the fundamental definition of extrinsic curvature of an hypersurface? Let me explain why I have not just copied one definition from the abundant literature. The specific structure on the Lorentzian manifold that I'm considering does not imply that an...
  50. PaulCam

    B Space Curvature & Potential Energy

    So I'm an Software Engineer, not a physicist, nor a mathematician. So I like to work in the qualitative, not the quantitative. Today I hit on a problem. I've been trying to remove the concept of "down" or "inward" from my thinking of gravity and GR. When people show the concept of space/time...
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