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I am learning differential geometry and have a few questions on curvature. -- Background:

  1. Gauss invented "Gauss curvature" to measure how surface curves.

  2. Riemann gives an ingenious generalization of Gauss curvature from surface to higher dimensional manifolds using the "Riemannian curvature tensor" (sectional curvature is exactly the Gauss curvature of the image of the "sectional" tangent 2-dimensional subspace under the exponential map).

  3. In modern textbooks on differential geometry, people usually first define the notion of connection, and then the Riemannian curvature tensor is expressed in terms of connection.

My questions are:

  1. Are there some other notions of "curvature", besides Gauss curvature and Riemannian curvature tensor as its generalization, which people have invented to measure how space curves?

  2. In history, who first introduced the notion of connection to describe the Riemannian curvature tensor, and why is this idea natural?

Thank you very much!

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    $\begingroup$ Question 1. Is clearly out of scope, except maybe if you consider Scalar and Ricci curvature to be part of Riemann curvature (which is more a tensor that leads to curvatures than a curvature by itslef). Even so, it needs little work to find generalizations of these (to graphs, metric spaces, measure-metric spaces, Markov chains) For Question 2 I guess the answer is Ehresman, but that needs checking. $\endgroup$ – Benoît Kloeckner Nov 29 '13 at 17:25
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    $\begingroup$ I think this is a good basic question. There are a couple votes to close but I'd like to suggest we leave this open. I think there are a few good answers already, and likely there will be more. $\endgroup$ – Ryan Budney Dec 1 '13 at 13:15
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    $\begingroup$ As I voted to close, I should probably explain more why. I think there may be a good question here (actually, two), but it seems like the OP has not done a thorough bibliographic search first. Also, more motivation would be welcome. That said, my vote is borderline and I don't see much harm if the question stays open. $\endgroup$ – Benoît Kloeckner Dec 1 '13 at 13:19
  • $\begingroup$ In the direction of Joseph's answer I believe there are a variety of notions of combinatorial curvature, where one is talking about triangulated manifolds, perhaps with some combinatorial analogue of a metric. It would be nice to have a few concrete examples of these. I don't know the literature well-enough myself to give anywhere near a complete answer to that. $\endgroup$ – Ryan Budney Dec 1 '13 at 13:34
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    $\begingroup$ This may not be what @RyanBudney had in mind, but: Robin Forman defines a notion of combinatorial curvature in his D&CG paper, "Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature" (Springer link). $\endgroup$ – Joseph O'Rourke Dec 1 '13 at 14:16

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in addition to these excellent examples of non-local curvature quantities and their extensions to the non-smooth setting (which I am not sure the OP was anticipating), I might add the 'original' non-local curvature measure: the holonomy.

Also, the OP was not precise about how to interpret the vague term 'space', so it is not at all clear how to answer the question about possible measurements of how 'space curves'. From the context of the question and the given description of the OP's background, I would guess that 'space' is intended to be interpreted as 'Riemannian manifold', in which case the OP is asking some fundamental questions that often aren't explicitly addressed in introductory expositions of Riemannian geometry: First, is the 'Riemann curvature tensor' the whole story in terms of understanding the local geometry of Riemannian manifolds (i.e., in describing how such 'spaces curve')? Second, how did the notion of a 'connection' arise historically, since Riemann didn't use it to define his measure of how 'space curves', and what was the motivation for introducing it? Both these questions frequently occur to beginners in the subject, so it's a bit surprising that they aren't treated a bit more explicitly in introductory books.

For example, consider the following situation: Suppose that, instead of Riemannian geometry on smooth manifolds, we were considering Hermitian geometry on complex manifolds, i.e., our manifolds are complex and we only consider Riemannian metrics that are complex compatible, in the sense that $g(iv) = g(v)$ for all tangent vectors $v$. In this case, the Riemannian curvature tensor of $g$ is not the whole story in (complex) dimensions bigger than $1$; in addition to the Riemann curvature tensor, one must consider the exterior derivative of the canonical $2$-form $\omega_g$ associated to $g$.

It is a nontrivial theorem that, in the case of Riemannian metrics, all of the 'pointwise' differential invariants of a metric $g$' (once these are properly defined) are generated by the 'Riemann curvature tensor' and its 'covariant derivatives'. (Note that this is stronger than what Riemann stated, which is that, if the Riemann curvature tensor vanishes, then the metric is locally Euclidean. It could have been, for example, that, when the Riemann curvature tensor vanishes identically, this overdetermined system of equations forces some other 'independent' invariants to vanish as well, due to integrability conditions of the overdetermined PDE system. As Weyl and Cartan proved, though, this does not happen in the case of Riemannian geometry.) [In the Hermitian case above, it turns out that the Riemann curvature tensor and $d\omega_g$, together with their 'covariant derivatives' suffice to generate all of the 'pointwise differential invariants of a Hermitian metric $g$ on a complex manifold'. This is also a nontrivial theorem.]

As far as how the notion of connection arose and why it is used so much in Riemannian geometry (and other geometries), that has an interesting history, and I will only sketch it here: In an 1869 paper, Elwin Christoffel showed that one could compute what we now call the Riemann curvature tensor of a metric $g$ as a differential expression in the coefficients of $g$ in two stages. First, one computes certain expressions, the 'Christoffel symbols' $\Gamma^k_{ij}$, in terms of the coefficients $g_{ij}$ of $g= g_{ij} dx^i dx^j$ and their first partial derivatives (with respect to the chosen local coordinate system), and then one computes the Riemann curvature tensor $R^i_{jkl}$ from the $\Gamma^i_{jk}$ and their first partial derivatives (with respect to the chosen local coordinate system). This gave an efficient method of computing the Riemann curvature tensor and exhibited it explicitly as a second order partial differential expression in the coefficients of $g$ in a somewhat manageable form.

Then, around 1890, Gregorio Ricci-Curbastro (for whom the Ricci tensor is named), realized that Christoffel's symbols could be used to define a notion of derivative for what we now call tensor fields in the presence of a background Riemannian metric $g$, one that would be independent of any choice of local coordinates, i.e., this derivative would be 'covariant', when one expresses both the tensor field and the metric in local coordinates. He and his former student, Tullio Levi-Civita, used this as the basis of their notion of 'absolute differential calculus'. This led to the idea of tensor fields 'parallel' (i.e., with vanishing absolute derivative) along curves and the notion of 'parallel transport', which gave a way of 'connecting' the tangent spaces along any (piecewise smooth) curve. This original notion of 'connection' (though I am not sure that Levi-Civita actually used this word as a noun) was then vastly generalized and applied to other geometries by Weyl, Schouten, and Cartan.

The general idea of curvature (and, indeed, holonomy) as a measure of how parallel transport depends on the choice of curve joining two points then became a fundamental notion and, in fact, it turns out that the Riemann curvature tensor of a metric $g$ is precisely the curvature (in this sense of dependence of parallel transport on the path) of the 'connection' introduced by Levi-Civita. [From this point of view, Weyl's theorem that the Riemann curvature tensor and its covariant derivatives give all of the 'pointwise differential invariants' of $g$ is by no means obvious, and it takes some serious work (with the appropriate definitions) to prove it.]

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    $\begingroup$ A very helpful answer indeed. By the way, I had asked a related question here. This is essentially asking about the last statement (Weyl's theorem) of your answer. $\endgroup$ – user90041 Dec 4 '13 at 3:59
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    $\begingroup$ Extremely nice and enlighting answer. Do you have some reference where the whole story is treated the way you present it, or does it come out of your experience (and then: why don't you turn it into a "reference"?). $\endgroup$ – Filippo Alberto Edoardo Sep 1 '14 at 21:46
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From outside the differential geometry literature, there has been a notion of cone-curvature used in shape recognition of surfaces in $\mathbb{R}^3$, e.g.:

"Extended cone-curvature based salient points detection and 3D model retrieval." Multimedia Tools and Applications. June 2013, Volume 64, Issue 3, pp 671-693. (Springer link)

The rough idea is to fit a cone at each point and use the cone's angle as an indicator of curvature. Below the points of high cone-curvature are more red:
   enter image description here
   (Image from Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications)

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There is a long history of curvature conditions known as "small cancellation conditions" which apply to group presentations, or if you prefer to the Cayley 2-complex of the presentation. These are combinatorial conditions on the structure of the presentation.

Gromov introduced several different measurements of curvature. One of them, now known as "Gromov hyperbolicity", is an asymptotic measurement that applies to path metric spaces, in particular to finitely generated groups via their Cayley graph; this property generalizes the properties of large triangles in the hyperbolic plane. Another one, actually one for each real number $K$ and known as $CAT(K)$, is a local measurement that generalizes properties of simply connected complete Riemannian manifolds whose sectional curvature is bounded above by $K$; as in ordinary Riemannian geometry, the cases $K<0$, $K=0$, $K>0$ have interesting theoretical differences.

Close to Ryan's comment and Joseph's answer, another curvature measurement introduced by Januskiewicz and Swiatkowski, and known simply as "simplicial nonpositive curvature" (the title of their paper), is a purely combinatorial condition that applies to simplicial complexes.

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An interesting "curvature" which has recently received much interest is the "Ma--Trudinger--Wang" (MTW) tensor, which arose in the study of when optimal transport maps are smooth on a Riemannian manifold. It can be thought of as a generalization of sectional curvature which depends in a nonlocal (!) way on the metric/distance function.

See http://cvgmt.sns.it/media/doc/paper/1148/Exp_1009_A_Figalli.pdf for a survey about the MTW tensor (it is defined in equation 5).

An interesting application of the MTW tensor to non-optimal transport questions in geometry is found in http://math.unice.fr/~rifford/Papiers_en_ligne/FRV-Sn.pdf, where its relationship to convexity of the cut locus and related topics are studied.

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The Gaussian curvature has a natural generalization to higher codimensional submanifolds of Euclidean space that for some reason has not appeared much (if at all) in the literature. One defines it as follows: Let $M \subset \mathbb{R}^k$ be an $m$-dimensional manifold, with $n=k-m$ the codimension. Take a point $p \in M$. For each unit vector $\nu \in \mathbb{R}^k$ normal to $M$ at $p$, we have a quadratic form $\text{II}_M^\nu(\cdot, \cdot) = < \text{II}_M(\cdot, \cdot), \nu >$. Define $K_M^\nu=\text{det}( \text{II}_M^{\nu} )/ \text{det}(\text{I}_M )$, where both determinants should be computed with respect to the same basis of $T_pM$. Let $NS_p \subset S^{k-1}$ denote the $(n-1)$-dimensional sphere of unit vectors normal to $M$ at $p$. If $\omega_d$ is the volume of the $d$-dimensional sphere, define the generalized Gaussian curvature of $M$ at $p$ to be \begin{equation} K_M = \frac{1}{\omega_{n-1} } \int_{NS_p} K_M^\nu \, dV_{NS_p}. \end{equation} You can check that for even-dimensional hypersurfaces this definition agrees with the classical definition. Other than the geometric interpretation of $K_M$ as an average of "directional Gaussian curvatures", the following two theorems justify the name give to $K_M$:

$\mathbf{Gauss-Bonnet\, \, Theorem}.$ \begin{equation} \int_M K_M \, dV_M = \frac{ \omega_{k-1} }{ \omega_{n-1} } \chi(M). \end{equation} $\mathbf{Theorema\, \, Egregium.\, \, }$ $ \omega_{n-1}/ \omega_{k-1} K_M$ is an intrinsic invariant of the metric.

In fact one has $\omega_{n-1}/ \omega_{k-1} K_M \, dV_M = \text{Pff}(-\Omega/2\pi)$ for even-dimensional manifolds. This is the best generalization of the Gaussian curvature as a SCALAR invariant that I know of.

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Following up on Ryan Budney's comment, here is Forman's definition of the combinatorial Ricci curvature of an edge (where {vertex, edge, face} = {0-cell, 1-cell, 2-cell}):


RicciCurvature
In words: The curvature of an edge $e$ of the manifold $\mathbb{M}$ is the number of vertices that are in the boundary of $e$, plus the number of faces whose boundaries include $e$, minus the number of edges that share only a vertex (and not a face) with $e$, minus the number of edges that share a face but not a vertex with $e$.

Note that the curvature of an edge $e$ under this definition is only dependent upon the 2-skeleton of $\mathbb{M}$. For a cube, Ric$(e)=2+2-0-2=2$:
 CubeEdge

The above definition is taken from Paul McCormick's thesis, "Combinatorial Curvature of Cellular Complexes" (PDF download), which in turn relies on Forman's paper, "Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature" (Springer link).

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  1. There are many things that are called "curvature". For example, Menger's curvature which plays an important role in Anaysis. Search "Menger curvature" anywhere in Mathscinet, to find out what is this.

  2. I suppose that Levi-Civita was the first to use connections, however I think that the term was introduced by E. Cartan.

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I want to answer your first question. The concept of symmetric curvature tensor is defined by N.Boroojerdian and A. Heidari. This concept is related to the notions divergence, Laplacian and Ricci Tensor. This curvature tensor is responsible for many behaviours of a geomteric space. see: "SYMMETRIC CURVATURE TENSOR,Bulletin of the Iranian Mathematical Society Vol. 37 No. 3 (2011), pp 249-267".

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Let $\gamma$ be a regular curve in the plan. we can assign various quantities $\tilde{\kappa}$ to $\gamma$ as follows: every quantity which is independent of parametrization, for example $\gamma^{(n)}.\gamma^{(m})/{\parallel \gamma' \parallel}^{n+m}$, etc... Such quantities are geometric invariants (independent of parametrization).

Now for a surface in $\mathbb{R}^{3}$, consider various normal sections to the surface. Denote the minimum and maximum values of the corresponding quantities by $\tilde{\kappa_{1}}$ and $\tilde{\kappa}_{2}$. It is interesting to find an algebraic operation on $\tilde{\kappa}_{1}$ and $\tilde{\kappa}_{2}$ (ex multiplication,...) such that the resulting quantity is an intrinsic number (invariant under isometry). Then generalize to $n$ dimensional objects with consideration of two dimensional sections.

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    $\begingroup$ Did anybody pursue this line of investigation? If yes, what interesting examples and results did they find? $\endgroup$ – Qfwfq Aug 25 '18 at 12:51
  • $\begingroup$ @Qfwfq To be honest I do not know any application or result in this line. $\endgroup$ – Ali Taghavi Aug 26 '18 at 7:28
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Assume that $M$ is a Riemannian manifold which is equipped with symplectic structure $\omega$.

Inspired by the definition of "Scalar curvature", one can define the quantity $tr_{\omega} Ric$ where $Ric$ is the Ricci curvature tensor associated to the metric but the trace is computed with respect to $\omega$ not with respect to the metric.

I am not sure whether this quantity is equal to the scalar curvature when $M$ is a Kahlar manifold.

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