Mathematics Stack Exchange
Mathematics Stack Exchange

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.

2039 questions
91
votes
4 answers

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
  • 1,727
65
votes
1 answer

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 -…
Selene Routley
  • 2,735
62
votes
3 answers

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…
Daniel Robert-Nicoud
  • 30,663
52
votes
3 answers

Do wheels of a car always travel the same distance?

Consider the left wheel and the right wheel of a car (say, rear wheels). The distances two wheels travel differ when they turn left or right. But, if the car starts traveling towards the north direction and ends traveling towards the same…
govindah
  • 822
41
votes
1 answer

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…
user166467
32
votes
3 answers

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
  • 32,819
29
votes
3 answers

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
  • 2,740
28
votes
2 answers

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…
Thomas Anton
  • 2,446
27
votes
3 answers

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
  • 928
27
votes
2 answers

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
  • 812
25
votes
4 answers

Are the Ricci and Scalar curvatures the only "interesting" tensors coming from the Riemannian curvature tensor?

In studying Riemannian geometry, one learns about the Riemann curvature tensor $$Rm(X,Y,Z,W) = \langle \nabla_X\nabla_YZ -\nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z, W\rangle$$ and its various symmetries. From the Riemann curvature tensor, one can define…
Jesse Madnick
  • 32,819
21
votes
3 answers

Relation between the 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
  • 1,269
21
votes
3 answers

Since the Curvature tensor depends on a connection (not metric), is it the relevant quantity to characterize the curvature of Riemannian manifolds?

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…
TheQuantumMan
  • 2,678
20
votes
3 answers

Why is the surface of a torus flat?

Why is the surface of a torus is said to be flat? If you consider the geometry of the torus, its surface has locally positive (spherical), negative (hyperbolic) and flat curvature.
Rene Kail
  • 309
19
votes
3 answers

What is a pullback of a metric, and how does it work?

The term "metric" is familiar, but not the idea of a pullback on it. I have tried to find intuitive, beginner-friendly explanations of this concept without success. Your attempts would be appreciated. Pictures and concrete examples would be…
Tensor McTensorstein
  • 455
1
2 3
…
99 100