Curvature Definition and 920 Threads

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. twistor

    Is the Weyl Curvature Hypothesis a Viable Alternative to Inflation?

    Hello guys. I was thinking about alternatives to inflation, especially old ones (such as the hawking-hartle state and imaginary time) and I remebered a theory put foward by Penrose, in which his relatively new CCC is based. Called the Weyl Curvature Hypothesis. No idea of what it is. Could you...
  2. S

    How to explain the curvature of space-time to students

    I teach a class on astronomy and recently tried to explain the curving of space time by massive objects like neutron stars and black holes. I even used a sheet of spandex to represent space-time which we bent using different weights. However my students were very confused how space, which they...
  3. FreeThinking

    What is equation for Lie derivative in Riemann curvature?

    Homework Statement (Self study.) Several sources give the following for the Riemann Curvature Tensor: The above is from Wikipedia. My question is what is \nabla_{[u,v]} ? Homework Equations [A,B] as general purpose commutator: AB-BA (where A & B are, possibly, non-commutative operators)...
  4. Breo

    Gauss - Bonnet Gravity -> Curvature variations

    So I am working on the next quadratic Lagrangian: $$ L = \alpha R_{\mu\nu}R^{\mu\nu} + \beta R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} + \gamma R² $$ I have already derived $$ \delta (R_{\mu\nu}R^{\mu\nu}) = [- \frac{1}{2}g_{\mu\nu}R_{\alpha\beta}R^{\alpha\beta} +...
  5. aditya ver.2.0

    Where should I ask about mathematical problems with Riemann curvature tensor

    I have come about few mathematical problems related to Riemann Tensor analysis while learning General Relativity. Should I ask these questions in this section or in the homework section. They are pretty hard!
  6. Shackleford

    Prove its Gauss curvature K = 1

    Homework Statement Assume that the surface has the first fundamental form as E = G = 4(1+u2+v2)-2 F = 0[/B] Homework Equations K = \frac{-1}{2\sqrt{EG}}[(\frac{E_v}{\sqrt{EG}})_v + (\frac{G_u}{\sqrt{EG}})_u][/B]The Attempt at a Solution Ev = -16v*(1+u2+v2)-3 Gu = -16u*(1+u2+v2)-3 When...
  7. C

    Is a Third-Rank Tensor Viable for Expressing Manifold Curvature?

    Hi, I am curious about expressing the curvature of a manifold using a third-rank tensor. The fourth order Riemann tensor can be contracted to give the second order Ricci tensor and zeroth order Ricci scalar, but is there no way of obtaining a third- or first-order tensors, or would this simply...
  8. V

    Reaction force due to the curvature and gradient drift

    We know that a charged particle will have a drift velocity in both a curved magnetic field and when there is a transverse spatial gradient in the magnitude of the magnetic field. This drift velocity is added to the rotation velocity around the the field line. In both cases the force vector on...
  9. N

    Finding radius of curvature of an eyeball

    Homework Statement Consider a simplified model of the human eye, in which all internal elements have the same refractive index of n = 1.40. Furthermore, assume that all refraction occurs at the cornea, whose vertex is 2.50 cm from the retina. Calculate the radius of curvature of the cornea such...
  10. T

    Cosmological Curvature: A $10 Bet with My Professor

    Good evening. I am a current astrophysics undergraduate who is currently having a $10 bet with my classical mechanics professor (chemistry/mathematics background) over what the official curvature of the universe is. While my academic level of understanding is still not quite high enough to...
  11. 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...
  12. 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...
  13. F

    Can Curvature Be Integrated Over Space in General Relativity?

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

    MHB What is the 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}}.
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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 $$R_{{abcd}} = - R_{{abdc}}$$...
  20. 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...
  21. 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...
  22. 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...
  23. 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?
  24. 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...
  25. 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...
  26. N

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

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  27. 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...
  28. 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...
  29. 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
  30. 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...
  31. P

    Dark energy and space time curvature

    How can we define SPACE TIME CURVATURE with respect to dark energy and dark matter ?
  32. 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...
  33. 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...
  34. 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...
  35. F

    Can Curvature Only Be Formed by Removing Sections of Flat 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...
  36. Phy_enthusiast

    Understanding Spacetime Curvature: Explained

    What does spacetime curvature means?
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. 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...
  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. W

    Is Mean Curvature Invariant Under Coordinate Changes?

    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...
  47. 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...
  48. 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?
  49. 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?
  50. 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...
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