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. C

    Question about Gravity and curvature of space time

    Hello all I just joined this forum so forgive me for jumping right in but I have a question about Gravity and the curvature of space time that I can't get answer with a Google search. My question: though I understand that an object remains in orbit because of the curvature of space time and it...
  2. N

    How to Verify Curvature Equation Using Chain Rule

    Homework Statement verify that |T ' (s)| = |T ' (t)| / |r ' (t)| Homework Equations K = |dT / ds| K = |r'(t) x r''(t)| / |r'(t)|^3 the x is a cross product The Attempt at a Solution I don't know how to start this problem because one side of the verification is in terms of arc length...
  3. F

    Is Curvature Accumulative?

    Is the curvature of GR accumulative? Can you integrate the curvature immediately around various points to find the curvature around a larger area? Thanks.
  4. D

    MHB Curvature of a graph at a point

    Consider the curve which is graph of a smooth function f : (a,b) → R. Show that at any {x}_{0}\:s.t\:{x}_{0} ∈ (a,b) the curvature is \frac{{f}^{''}({x}_{0})}{{(1+{{f}^{'}({x}_{0})}^{2})}^{3/2}}.
  5. lovexmango

    What is the parametrization of the graph of ln(x)?

    Let k(x) be the curvature of y=ln(x) at x. Find the limit as x approaches to the positive infinity of k(x). At what point does the curve have maximum curvature? You're supposed to parametrize the graph of ln(x), which I found to be x(t)=(t,ln(t)). And you're not allowed to use the formula with...
  6. C

    Riemannian curvature of maximally symmetric spaces

    A maximally symmetric is a Riemannian n-dimensional manifold for which there is n/2 (n+1) linearly independent (as solutions) killing vectors. It is well known that in such a space $$R_{abcd} \propto (g_{ab}g_{cd} - g_{ac}g_{bd}) .$$ How is this formula derived for a general maximally...
  7. A

    MHB Why curvature of a plane curve is k =d(phi)/ds?

    Why curvature of a plane curve is k = \frac{d\phi }{ds} ? I know that curvature of a plane is \frac{\left | r'(t) \times r''(t) \right |}{\left | r'(t) \right |^3} , and that led to this. k = \frac{\left | \frac{d^2s}{dt^2} \right |sin\phi }{\left | \frac{ds}{dt}\right |^2} But I can't go...
  8. C

    Intuition on the Friedman equation: curvature and expansion

    The Friedmann equation states that $$(\frac{\dot a}{a}) = \frac{8\pi G}{3} \dot \rho + \frac{1}3 \Lambda - \frac{K}{a^2},$$ where ##a, \rho, \Lambda, K## respectively denotes the scale factor, matter density, cosmological constant and curvature. Now, I'm trying to get at an intuition on...
  9. E

    Riemann Curvature Tensor Symmetries Proof

    I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies $$R_{{abcd}} = - R_{{bacd}}$$ Antisymmetry last pair of indicies $$R_{{abcd}} = - R_{{abdc}}$$...
  10. S

    Curvature of Light Paths Near a Mass

    If I understand everything correctly, space near (but outside) a mass is curved negatively, so that if I create a triangle with, for example, rigid rods and the mass in its center, the angles would sum up to less than 180°. (If I am mistaken, please correct me.) On the other hand, the typical...
  11. W

    What is Active Curvature Mass?

    I’ve just read Schutz: “Gravitation from ground up”. He says that there are 4 (not all independent) sources of gravity: 1. density of active gravitational mass = (rho + 3 P) 2. active curvature mass (generating spatial curvature) = (rho - P/c^2) 3. ordinary momentum...
  12. L

    Why does spatial curvature become observable only in late universe?

    So the universe starts with an amount of matter, radiation, a fixed spatial curvature constant and a cosmic constant. Due to the expansion of the universe matter and radiation dillute and the spatial curtvature decreases whereas the cosmic constant remains fixed. Radiation dillutes faster...
  13. Shackleford

    Calculating Curvature of Non-Unit-Speed Curve Using a Trig Identity

    1. Compute the curvature. α(t) = (cos^3t, sin^3t) This is not a unit-speed curve. I want to use κ(t) = \frac{||T'(t)||}{||σ'(t)||} When I find α'(t) and then its norm, I run into an impasse. Am I supposed to use a trig identity?
  14. S

    Is curvature possible for a 2D metric?

    I was recently trying to test something out with the Riemann tensor. I used only 2 dimensions for simplicity sake. As I was deriving the Riemann tensor, I noticed that it looked as if all of the elements were going to come out to be 0 (which they all did). Therefore, this coordinate system is...
  15. S

    Low curvature effective action in string theory

    String effective action: S=-\frac{1}{2\lambda_{\text{s}}^{d-1}}\int d^{d+1}x\sqrt{|g|}e^{-\phi}\left[R+(\nabla\phi)^2+2\lambda_{\text{s}}^{d-1}V(\phi)-\frac{1}{12}H^2\right]+S_m where H^2=H_{\mu\nu\alpha}H^{\mu\nu\alpha}\\ H_{\mu\nu\alpha}=\partial_\mu B_{\nu\alpha}+\partial_\nu...
  16. N

    If two lenses have the same radii of curvature but different indexes

    Homework Statement If two lenses have the same radii of curvature but different indexes of refraction their focal lengths won't be equal because the one with the greater index of refraction will undergo greater refraction and will have a smaller f. Doesn't this contradict the definition of...
  17. S

    Questions about the Riemann Tensor

    We know how the curvature of a vector V or a manifold is depicted by the following formula: dx\mudx\nu[∇\nu , ∇\mu]V Now we know that the commutator is simply the Riemann tensor. My question here is: How do you actually apply that vector V to the Riemann tensor? Here is an example of what I...
  18. D

    Radius of curvature of second lens surface

    Homework Statement A lens of power of -5.0D has a surface which is convex of radius of curvature of 15.0cm. The lens is made of material of refractive index of 1.50. What's the radius of the other surface of lens? Homework Equations The Attempt at a Solution since power = 1/f...
  19. D

    Radius of curvature of glass and water lens

    Homework Statement by taking the lower curvature as r1 , and the upper curvature as r2 , i don't know whether r1 is 20cm , r2 is 10cm or vice versa. But according to the ans r1= 10 cm , r2= 20cm . why is it so? Homework Equations The Attempt at a Solution
  20. D

    Radius of curvature of partially cut glass surface

    Homework Statement when the glass is partially cut( as shown in the photo ) , the centre of curvature is inside the denser medium (glass), so the centre of curvature should be lower than point Q in the diagram . am i correct? by saying that the centre of curvature is inside the denser medium...
  21. P

    Dark energy and space time curvature

    How can we define SPACE TIME CURVATURE with respect to dark energy and dark matter ?
  22. J

    Dark Matter. Space-Time curvature. Galaxy formation

    1. Gravity is the geometric curvature of space-time caused by massive objects. 2. Dark Matter surrounds galaxies. 3. Dark Matter is thought to be critical in galaxy formation. 4. The mass of Dark Matter creates curvatures in space-time around baryonic matter which forms galaxies. What roles...
  23. M

    Riemann curvature of a unit sphere

    The Riemann curvature of a unit sphere is sine-squared theta, where theta is the usual azimuthal angle in spherical co-ordinates, and this is shown in many textbooks. But since a sphere is completely specified by its radius, then as far as I can see its curvature should be a function of its...
  24. C

    Factor 1/2 in the Curvature Two-form of a Connection Principal Bundle

    In the formulation of connections on principal bundles, one derives an expression for the covariant exterior derivative of lie-algebra valued forms which is given by $$D\alpha = d \alpha + \rho(\omega) \wedge \alpha,$$ where ##\rho: \mathfrak g \to \mathfrak{gl}(\mathfrak g)## is a...
  25. F

    Curvature from holes in space?

    I wonder if curvature necessarily means space has been removed. The typical example is forming a "curved" surface by cutting out a triangle from a flat surface, and then gluing the remaining side back together. This forms of a cone which is a type of curved surface. What is the generalization of...
  26. Phy_enthusiast

    Understanding Spacetime Curvature: Explained

    What does spacetime curvature means?
  27. Q

    Curvature 1-forms in NP formalism

    Hey guys, I'm working on a summer research project right now in diff. geo. I'm at the point where I have to define the spin coefficients for my spacetime. I'm following an appendix in another paper related to my problem (the equivalence problem for 3D Lorentzian spacetimes). In the appendix I...
  28. M

    Radius of Curvature: Formula & Name

    I am working on a paper that provides the following formula for computing radius of curvature at a point on a surface. \frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E}G} where E,G are first fundamental coefficients and S is the arc length parameter. Can anyone please tell me the...
  29. S

    Question about Riemann and Ricci Curvature Tensors

    After my studies of metric tensors and Cristoffel symbols, I decided to move on to the Riemann tensor and the Ricci curvature tensor. Now I noticed that the Einstein Field Equations contain the Ricci curvature tensor (R\mu\nu). Some sources say that you can derive this tensor by simply...
  30. DiracPool

    3-d versus 4-d spacetime curvature

    A second SR question that has been on my mind lately is that of hyperbolic nature of Minkowski space. The fact that the invariant interval, or lines of constant delta S trace out a hyperbola according to the equation, ##x^2-(ct)^2=S^2##, is fascinating to me and seems to imply that space-time...
  31. Drakkith

    Types" of Space-Time Curvature in GR

    When we talk about space-time curvature or the curvature of space, how many different "types" of curvature are there according to GR? For example, the rounded surface of a cylinder is curved in only 1 dimension, while the other is flat. For a sphere, both dimensions of the surface are curved...
  32. M

    Geodesic Radius of Curvature Calculation Method

    I am trying to compute the geodesic (or tangent) radius of curvature of the geodesic circle by using the below formula. \frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G} where s is the arc length parameter and E, G are the coefficents of the first fundamental form. Can you...
  33. WannabeNewton

    2-point correlation functions for curvature perturbations

    Does anyone have a good reference or references that go into detail on rigorous/formal developments of 2-point correlation functions for curvature perturbations (and related perturbations) in the cosmological context? I'm using the TASI lectures in inflation, Mukhanov, and Dodelson but none of...
  34. M

    Relation between unit tangent/normal vectors, curvature, and Lin. Alg.

    Hey there, This isn't a homework question, it's for deeper understanding. So I'm learning about unit normal/tangent vectors and the curvature of a curve. I have a few questions/points. 1) So my book states that we can express acceleration as a linear combination of the acceleration in the...
  35. DreamWeaver

    MHB A curvature problem (differentiation)

    In the Euclidean plane, assume a differentiable function y=f(x) exists. At any given point, say (x_0,y_0), the line tangential to y=f(x) at this point intersects the x-axis at an angle \phi. The curvature of this curve, \kappa, is the rate of change of \phi with respect to arc length, s...
  36. W

    Mean Curvature and Invariance.

    Hi All: I am curious about the definition of mean curvature and its apparent lack of invariance under changes of coordinates: AFAIK, mean curvature is defined as the trace of the second fundamental form II(a,b). II(a,b) is a quadratic/bilinear form, and I do not see how its trace is invariant...
  37. M

    Curvature at t=0 for r(t) = 4/9(1+t)^(3/2)i + 4/9(1-t)^(3/2)j + 1/3t k

    Homework Statement Find the curvature of ##r(t) = \frac 4 9 (1+t)^ \frac 3 2 i + \frac 4 9 (1-t)^ \frac 3 2 j + \frac 1 3 t \hat k## at t=0 Homework Equations K=1/|v| * |dT/dt| The Attempt at a Solution Found v. ##v= \frac 2 3 (1+t)^ \frac 1 2 i - \frac 2 3 (1-t)^ \frac 1 2 j + 1/3 \hat...
  38. ChrisVer

    Changing curvature of a manifold

    Is it possible for a riemann manifold to change its curvature? In practice could the universe in general change its curvature by time? (let's say in the past it was negative and today it's almost flat tending to positive); If not which theorem disproves it?
  39. HowardHughes

    Imagining spacetime curvature more accurately

    I am intrigued to see what spacetime curvature is like in reality. Most images or ways to imagine it tend to look at spacetime as a fabric which it is not precisely. So how would be best to imagine it... Do any of the picture demonstrate this? What is the best way to imagine it?
  40. anorlunda

    Must the curvature of space be constant?

    I was listening to one of Leonard Susskind's cosmology lectures. He talks about the factor K having values of +1/0/-1 corresponding to positive/flat/negative curvature. We don't know what the real value of K is. But then as he discussed K at the big bang and K now, it seemed that he was...
  41. T

    Exploring the Limits of Modeling EM Forces as Spatial Curvature

    Why can't EM attraction/repulsion be modeled as spatial curvature the way gravity can be? And for that matter, why can't the strong and weak nuclear forces be modeled that way either? Or can they?
  42. A

    Radius of curvature R of alpha particle beam

    Homework Statement Homework Equations r=mv/qb mv=sqrt(2*KE*m(alpha)) m(alpha)=6.64e-27 kg The Attempt at a Solution i was just wondering how to get the answer (7.6e-4 m). i get path curving down, and do r=sqrt(2*1e3eV*6.64e-27kg*1.6e-19J/eV)/(q*B) =15.18e-4m so to get the...
  43. S

    Light bending in the hot plate model of curvature

    Light bending in the "hot plate" model of curvature In the Feynman lectures, feynman describes the hot plate model of space curvature and shows that light is bent around the center of the plate, see Fig. 42-6 http://www.feynmanlectures.caltech.edu/II_42.html#Ch42-S1 However, the hot plate...
  44. Z

    Radius of curvature of planetary motion

    Suppose we have a planet of mass m orbiting a larger one of mass M along an elliptical path. If we use polar coordinates with the origin placed on the planet of mass M (focus of the ellipse) then at the instant when the smaller planet is at the point of closest approach we have: \boldsymbol{v}...
  45. B

    Curvature of a circle approaches zero as radius goes to infinity

    Hello, this isn't a homework problem, so I'm hoping it's okay to post here. I would like to know the correct way to mathematically express the idea in my title. It is intuitively obvious that as the radius of a circle increases, it's curvature decreases. I looked it up and found that...
  46. R

    If gravity is curvature of spacetime, why unified force @ Planck epoch

    Hi Bear with my possible ignorant. I am puzzled over this dilemma. If General Relativity states that gravity is the curvature of spacetime, that is, no spacetime no gravity, and the cause of curvature is matter (mass), it means that if no matter, there is no gravity. I understand that...
  47. C

    Cushion seat angle : Radius of curvature

    The problem is stated in the attachment. I would include my attempt at the question if I got anywhere. I'm really only looking for a hint as to how I set up the solution. PS, I understand how to work out the angle if the car wasn't moving. Thanks Homework Statement...
  48. A

    How Can Software Identify Maximum Curvature on a Graph?

    Urgent!How to find the point where Maximum Curvature occurs on graph? Homework Statement The graph(2D) is plotted from experimental data, not from a given equation and within certain limit. I want to know the x&y coordinate where the maximum curvature occur via software means, not manually...
  49. 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...
  50. 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...
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