Questions tagged [curvature]

In differential geometry, the term curvature tensor may refer to the Riemann curvature tensor of a Riemannian manifold, the curvature of an affine connection or covariant derivative (on tensors), or the curvature form of an Ehresmann connection. (Def: http://en.m.wikipedia.org/wiki/Curvature_tensor)

In differential geometry, the term curvature tensor may refer to the Riemann curvature tensor of a Riemannian manifold, the curvature of an affine connection or covariant derivative (on tensors), or the curvature form of an Ehresmann connection. Reference: Wikipedia.

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds. Reference: Wikipedia.

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Is there any easy way to understand the definition of Gaussian Curvature?

I am new to differential geometry and I am trying to understand Gaussian curvature. The definitions found at Wikipedia and Wolfram sites are too mathematical. Is there any intuitive way to understand Gaussian curvature?
Shan
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Geometrical interpretation of Ricci curvature

I see the scalar curvature $R$ as an indicator of how a manifold curves locally (the easiest example is for a $2$-dimensional manifold $M$, where the $R=0$ in a point means that it is flat there, $R>0$ that it makes like a hill and $R<0$ that it is…
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Are there simple examples of Riemannian manifolds with zero curvature and nonzero torsion

I am trying to grasp the Riemann curvature tensor, the torsion tensor and their relationship. In particular, I'm interested in necessary and sufficient conditions for local isometry with Euclidean space (I'm talking about isometry of an open set -…
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Intuition for curvature in Riemannian geometry

Studying the various notion of curvature, I have not been able to get the intuition and deeper understanding beyond their definitions. Let me first give the definitions I know. Throughout, I will consider a $m-$dimensional Riemannian manifold…
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Geometric interpretation of connection forms, torsion forms, curvature forms, etc

I have just begun learning about the connection 1-forms, torsion 2-forms, and curvature 2-forms in the context of Riemannian manifolds. However, I am finding it hard to relate these notions to any sort of geometric intuition. How can one interpret…
Jesse Madnick
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Is there a Stokes theorem for covariant derivatives?

A $V$-valued differential $n$-form $\omega$ on a manifold $M$ is a section of the bundle $\Lambda^n (T^*M) \otimes V$. (That is, the restriction $\omega_p$ to any tangent space $T_p M$ for $p \in M$ is a completely antisymmetric map $\omega_p : T_p…
Turion
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Is there a geometric explanation for why principal curvature directions are orthogonal?

Let $f: \mathbb{R}^2 \supset M \rightarrow \mathbb{R}^3$ be a smooth immersion and let $N: M \rightarrow S^2 \subset \mathbb{R}^3$ be the corresponding Gauss map. The normal curvature along a unit tangent direction $X \in TM$ can be expressed as $$…
PolyKnowMeAll
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Why should I care about Gauss-Bonnet (and Gaussian curvature)?

The Gauss-Bonnet Theorem (for orientable surfaces without boundary) states that for surface $M$, with Gaussian curvature at a point $K$, we have $$\int_M K\ dA=2\pi\chi(M).$$ Right now, this just says to me that the integral of something I don’t…
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What comes after the curvature tensor in "higher derivatives"?

Let $(M,g)$ be a Riemannian Manifold and $\nabla$ be a metric compatible (not necessarily Levi-Civita) connection on $M$. The metric tensor itself is the 0-th order term in the covariant derivatives wrt to connection. The next order is the torsion…
Gonenc
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Relation bewteen Hessian Matrix and Curvature

According to Hessian matrix, It describes the local curvature of a function. AFAIK, for one-variable function $f(x)$, its local curvature is $$\kappa = \frac{|f''|}{(1 + f'^2)^{3/2}},$$ and its Hessian matrix is $$\mathcal{Hess}(f) =…
avocado
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Total rotation of a circle (or other closed curve) when 'rolled' along a curve in $\mathbb{R}^2$

As to compute how much a circle rotates when 'rolled' along a curve in $\mathbb{R}^2$, the most intuitive way to me to find the number of rotations is: $S/C+W/(2\pi)$ $S$ is the arc-length of the curve $C$ is the circumference of the circle $W$ is…
MeMyselfI
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How to prove a manifold is simply connected?... using geometry

I was Looking at another questions title, and given the tag of DG, I thought it would read a little more like this one. Or at least that answers to this question would be answers to that question. There are many different techniques one might use to…
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Direct proof of the second Bianchi identity

Let $X$,$Y$,$Z$,$W$ be vector fields on a riemannian manifold, and let $R(X,Y)W$ be the riemannian curvature: $$ R(X,Y)W = \nabla_{X}\nabla_{Y}W - \nabla_{Y}\nabla_{X}W - \nabla_{[X,Y]}W $$ Let $g$ be the metric tensor and $\tau$ be the torsion…
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Hyperbolic diameter of Amsler's surface

I've recently learned about Amsler's surface, a surface of constant negative Gaussian curvature. If I understand things correctly, there is a whole family of such surfaces, differing in the angle of intersection for the two lines that generate it.…
MvG
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Since the Curvature tensor depends on a connection (not metric), is it the relevant quantity to characterize the curvature of Riemannian manifolds?

1) The definition of the Riemann curvature tensor does not include a metric. So, if we have a smooth manifold(not a Riemannian manifold), we can define the Riemannian curvature tensor for it by just giving it a connection(not the Levi-Civita…
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