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. A

    I What is the source of curvature in an accelerating frame?

    I can understand that a free falling frame in a curved space with a non-zero Riemann tensor has a zero Ricci tensor but I have a doubt about the opposite; does an accelerating frame in a vacuum space have a non-zero Ricci or Riemann tensor? if so, where do components of energy momentum tensor...
  2. arupel

    B Is Space Infinite? Examining the Big Bang Theory and the Curvature of Spacetime

    From what I read attempts to measure the curvature of space have not succeeded. It would seem there may not be a curvature of space time. If this is true then what may be implied is that space goes on forever. If this is true how could the big bang theory, if it could, give a reasonable answer...
  3. darida

    Derivative of Mean Curvature and Scalar field

    Homework Statement Page 16 (attached file) \frac{dH}{dt}|_{t=0} = Δ_{Σ}φ + Ric (ν,ν)φ+|A|^{2}φ \frac{d}{dt}(dσ_{t})|_{t=0} = - φHdσ H = mean curvature of surface Σ A = the second fundamental of Σ ν = the unit normal vector field along Σ φ = the scalar field on three manifold M φ∈C^{∞}(Σ)...
  4. M

    How does curvature of an arch bridge affect its strength?

    Hey, I've been looking into some civil/structural engineering for a school project, and came across bridge design. I've decided to try some integrating and optimising to do with an arch bridge (optimal cost/strength proportions). The math isn't too hard, but what I'm struggling with so far is...
  5. Elnur Hajiyev

    A Can geodesic deviation be zero while curvature tensor is not

    I understand(or assume understand) that geodesic deviation describes how much parallel geodesics diverge/converge on manifolds while moving along these geodesic. But is not it a definition for intrinsic curvature? If it is same as Riemann curvature tensor in terms of describing curvature, why...
  6. F

    I Is Gravity inertia, acceleration or curvature in GR?

    Hello Forum, I have read Einstein famous thought experiments about the elevator. 1) Being inside an elevator accelerating upward in absence of a gravitational field is equivalent to being inside the same elevator at rest inside a homogeneous gravitational field. 2) An elevator in free fall...
  7. P

    B Gaussian Curvature and Riemmanian Geometry

    Please bear with me because I'm only in Pre-calculus and am taking basic high school physics. This is completely outside of my realm but curiosity has taken the better of me. I just learned last week about the difference between Euclidean Geometry and Riemmanian Geometry (from another thread...
  8. R

    Are Gravitational Waves and Waves Transmitting Curvature Changes Different?

    G-Waves is a buzzword recently :) At the beginning I thought G-waves as the propagation of the changes of the curvature caused by a mass when the status of the mass (e.g. value or location) changes...But moment ago, I was told that G-waves are different from the waves that transmitting the...
  9. W

    Curvature effect can be neglected

    Homework Statement why when ℓ / R ≪ 1 , the curvature effect can be neglected ?? Homework EquationsThe Attempt at a Solution no matter how small or how big is R , it is strill considered as curvature , right ??
  10. marcus

    I "Dark energy" may be nothing more than baseline curvature

    I know of no scientific reason to suppose that "dark energy" is anything more than the cosmological curvature constant identified by Einstein in 1917 as occurring naturally in the GR equation for spacetime curvature. It might eventually turn out to be related to some type of energy. That's...
  11. W

    Berry's Curvature Equation cross product calculation

    Hi, The following textbook Heisenberg's Quantum Mechanics shows an example of calculating Berry's curvature (top page on pg 518). It led to a following equation Vm= (- 1/B2 ) * i *∑ ( <m,B|S|n,B> ∧ <n,B|S|m,B> ) / A2 ...[1] the textbook claims that we add the term m = n since <m|S|m> ∧ <m|S|m>...
  12. Kevin McHugh

    Solve Reimann Curvature: Diff Geom Tips for Computing 256 Components

    Should I post in Diff Geometry? I searched that forum, and did not see what I was looking for. I want to compute all 256 components of the curvature tensor. Do I start with the equation of geodesic deviation in component form, or can I go straight to the definition of the components in terms of...
  13. lynchmob72

    How does space-time curvature affect light?

    If space is warped around heavy objects in space, i feel that space would be FUBAR around black holes. So, my question is, Does light get sucked in by gravity, or does it just get caught in the warped space around a black hole?
  14. J

    Curvature of a group manifold

    So I'm working with some group manifolds. The part that's getting to me is the Ricci scalar I'm using to describe the curvature. I have identified the groups that I'm using but that's not really relevant at the moment. We're using a left-invariant metric ##\mathcal{M}_{ab}##. Now I've got the...
  15. W

    Curvature at the origin of a space as described by a metric

    Homework Statement This is a problem from A. Zee's book EInstein Gravity in a Nutshell, problem I.5.5 Consider the metric ##ds^2 = dr^2 + (rh(r))^2dθ^2## with θ and θ + 2π identified. For h(r) = 1, this is flat space. Let h(0) = 1. Show that the curvature at the origin is positive or negative...
  16. C

    Find Tangential Component of Acceleration and Curvature

    Homework Statement Homework Equations I know that the tangential accel is v = wr and that Centripetal = v^2/r The Attempt at a Solution For A, I thought it would be straight forward if I had the radius as well as omega. I know that the distance between A and B is 60in, but I don't think it...
  17. F

    Can a "density" of points create curvature?

    Spacetime shrinks in a gravitational field. As I understand it, objects falling into a black hole will appear to contract in size and run slower as they approach the horizon. This is similar to how things contract and slow down when traveling close to the speed of light because they are...
  18. Pianorak

    Inflation and Curvature: Examining the Relationship Since the Big Bang

    Since the very rapid expansion of the Universe, ie Inflation, caused curvature to be smoothed out, the assumption must be that curvature existed between the time of the Big Bang and the start of Inflation. If that is correct, is this borne out by the Planck or any other theory?
  19. bcrowell

    Eigenvalues of curvature tensors as curvature scalars?

    I've been playing around with the Carminati-McLenaghan invariants https://en.wikipedia.org/wiki/Carminati–McLenaghan_invariants , which are a set of curvature scalars based on the Riemann tensor (not depending on its derivatives). In general, we want curvature scalars to be scalars that are...
  20. Omega0

    Curvature Numerical Gauss etc.

    Hi! I would like to calculate the curvature for a surface S:R^2->R'3 numerically. The problem: I simply have the surface as a mesh like you see in the image attached. I calculated the linearly interpolated face normals in the nodes, too. You see the vectors. Question: How would you calculate...
  21. bcrowell

    Curvature polynomials vanish for plane waves?

    Geroch 1968 touches on the Kundt type I and II curvature invariants. If I'm understanding correctly, then type I means curvature polynomials. Type II appears to be something else that I confess I don't understand very well. (I happen to own a copy of the book in which the Kundt paper appeared. I...
  22. O

    Refraction as an explanation for light curvature

    Wikipedia states that: "If the measurement is close enough to the surface, light rays can curve downward at a rate equal to the mean curvature of the Earth's surface. In this case, the two effects of assumed curvature and refraction could cancel each other out and the Earth will appear flat in...
  23. M

    Finding Curvature of Function

    Homework Statement For the first problem I am asked to find the curvature for y=cosx We are studying vector value functions so I tried to rewrite this as a vector valued function so I can find the curvature. I just chose r(t)= <t,cost,0>. I found rI(t)=<1,-sint,0> and rII(t)=<0,-cost,0> and...
  24. J

    Example of curvature scalar diverging at infinity?

    Reading Geroch's "What is a Singularity in General Relativity?", it seems that polynomial scalar invariants constructed from the Riemann tensor can diverge if we are at infinite distance, and not in a true singularity. Can someone give an example of space-time whose scalar invariant diverges...
  25. bcrowell

    Curvature singularity with well-behaved Kretschmann scalar

    Does anyone know of an example, preferably a simple one, that can be used to demonstrate that we can have a curvature singularity without a singularity in the Kretschmann scalar?
  26. sunrah

    Why is the non-zero value of spatial curvature +/- 1?

    Going from the Newtonian to relativistic version of Friedmann's equation we use the substitution kc^{2} = -\frac{2U}{x^{2}} The derivation considers the equation of motion of a particle with classic Newtonian dynamics. I can sought of see that if space is flat the radius of curvature will be...
  27. K

    Space-Time Curvature in General Relativity

    Suppose we are in a Minkowskian space, away from all the source of gravity, and observe an accelerated frame from this frame. Acoording to Equivalence principle, we can consider the accelerated frame to be at rest and assume we have gravity in the accelerated frame. Thus, observer in the...
  28. G

    Engineering a body with an intrinsic curvature

    I don't know much about differential geometry but I hope this is a good place to ask and that my question "makes sense" I have heard that an ornamental cabbage leaf is an example of a surface with an intrinsic curvature. If one wanted to make such a surface from scratch(and to detailed...
  29. W

    Can curvature ever be greater than at relative extremum?

    Homework Statement For a generic function y=f(x) which is twice-differentiaable, is it possible for there to be a curvature on the curve of that function that is greater than the curvature at its relative extremum?Homework Equations The Attempt at a Solution From visualization and a sketch...
  30. W

    Finding the Limit of Curvature for a Polar Curve

    Homework Statement Given the polar curve r=e^(a*theta), a>0, find the curvature K and determine the limit of K as (a) theta approaches infinity and (b) as a approaches infinity. Homework Equations x=r*cos(theta) y=r*sin(theta) K=|x'y''-y'x''|/[(x')^2 + (y')^2]^(3/2) The Attempt at a Solution...
  31. TyroneTheDino

    Curvature and Tangent Line Distance Relationship

    Homework Statement Let T be the tangent line at the point P(x,y) to the graph of the curve ##x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}, a>0##. Show that the radius of curvature at P is three times the distance from the origin to the tangent line T.Homework Equations R=1/K ##R=\frac{\left...
  32. TyroneTheDino

    Curvature and radius of curvature of a cartesian equation

    Homework Statement A highway has an exit ramp that beings at the origin of a coordinate system and follows the curve ##y=\frac{1}{32}x^{\frac{5}{2}}## to the point (4,1). Then it take on a circular path whose curvature is that given bt the curve ##y=\frac{1}{32}x^{\frac{5}{2}}## at the point...
  33. C

    MHB How to Find the Curvature of r(t)?

    Find the curvature. r(t) = 9t i + 5 sin(t) j + 5 cos(t) k
  34. C

    MHB Find the curvature at the point

    Find the curvature at the point (2, 4, −1). x = 2t, y = 4t3/2, z = −t2
  35. C

    MHB Find the curvature of the vector

    Find the curvature. r(t) = 3t i + 5t j + (6 + t2) k κ(t) = This is what i have so far: derivative of r(t) = 3i + 5j + 2k I know the next step is to find the cross product of r(t) and r'(t) but I'm not sure how to go about it, especially how to make use of the k vector of r(t)
  36. DameLight

    What is the Curvature of a Line in Calc III?

    Hi, I am taking Calc III, but I am having a hard time understanding some of the concepts. Right now I am struggling with understanding the curvature of a line. What I have in my notes is this: Curvature second derivative (rate of change of tangent line)(rate of change w/ respect to arc length)...
  37. D

    Deriving Riemann Tensor Comp. in General Frame

    How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection? I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...
  38. T

    Heory of Spacetime Curvature and Gravity

    Do gravitational forces have to follow spacetime in the same way as light? Or does gravity act in a higher dimension? Thanks, T
  39. H

    Exploring Physics: A Newbie's Questions on Space and Curvature

    Hello, I just joined. I have no formal background in physics, just curiosity. So, my questions may well be simplistic to most of you. Hopefully, that is permissible, now and then! First question, if space, as it's usually defined, is empty, nothing, how can it be curved? Hankb
  40. P

    What is a nonscalar curvature singularity?

    What is a nonscalar curvature singularity, in the context of "the https://en.wikipedia.org/w/index.php?title=Wave_of_death&action=edit&redlink=1 is a gravitational plane wave exhibiting a strong nonscalar null...
  41. J

    Definition for curvature of worldline in Minkowski space

    Hello, in this section of the wiki article on Rindler coordinates it is stated that the proper acceleration for an observer undergoing hyperbolic motion is just "the path curvature of the corresponding world line" and thus a nice analogy between the radii of a family of concentric circles and...
  42. newjerseyrunner

    Over what scale is curvature measurable

    Imagine I have three space probes that I send out radially. They have a superluminal way to determine each other's relative position to each other instantaneously. If each one measures the relative position of the other two and comes up with an angle for them, how far away would they have to...
  43. Tony Stark

    Curvature of spacetime inside hollow sphere

    If mass curves spacetime in its vicinity, then consider the following case- Take a heavy hollow lead sphere which has 2 smaller lead balls placed in it. The Outer Sphere will curve spacetime around itself and thus will have its own gravity, but what about the 2 balls placed in it? The spacetime...
  44. P

    Find the radius of curvature of a particle

    Homework Statement A particle is moving on a path parameterized as such: $$x(t)=a\sinωt \quad y(t)=b\cosωt$$ Find the radius of curvature ρ as a function of time. Give your answer in Cartesian coordinates. Homework Equations $$\frac{1} {Radius~of~curvature}=|\frac{de_t}{ds}| $$, where et is...
  45. Chronos

    Is the Universe Flatter Than We Thought?

    This paper;http://arxiv.org/abs/1508.02469 ,Geometrical Constraint on Curvature with BAO experiments, suggests an improvement of curvature constraints on the geometry of the universe. On a purely geometric basis the authors' measurements suggest σ(ΩK) \simeq0.006 The GR+Λ case yields a value...
  46. P

    Checking derivation of the curvature tensor

    Homework Statement I am trying to derive the curvature tensor by finding the commutator of two covariant derivatives. I think I've got it, but my head is spinning with Nablas and indices. Would anyone be willing to check my work? Thanks Homework Equations I am trying to derive the curvature...
  47. jfizzix

    Curvature implying Closedness in N dimensions

    A two-dimensional surface with everywhere positive curvature is a closed surface with no boundary (isomprphic to a sphere). Is this true for higher dimensional surfaces as well? Would a three-dimensional surface, with everywhere positive curvature be a closed hypersurface isomorphic to a...
  48. A

    I Question about gravity and speed of light

    Hi all, After reading about Einstein's theory of relativity I have few questions as follows 1. Let's say I am in a space lab which is traveling at the speed of half of the speed of light. So when I try to measure the speed of light coming from space I record it as 'c'. Time is running slow...
  49. ORF

    Spacetime Curvature: Does It Affect Objects?

    Hello Does the spacetime curvature produced by an object affect the object itself? Thank you in advance :) Greetings!
  50. S

    Gravity & Spacetime Curvature: Have I Understood?

    When spacetime is not bent the two objects, red ball and blue ball, will move strait up the y-axis as they move through time. (Space is x and time is y). Now I've made the assumption that either a) All things want to move the smallest possible distance to the next point in time or b) all...
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