Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.

learn more… | top users | synonyms

1
vote
0answers
83 views

Reference request: Ebin

I'm after the paper The manifold of Riemanian metrics by D. Ebin. A link to the reference is: http://www.ams.org/mathscinet-getitem?mr=0267604 The paper seems to be very hard to track down. Can ...
-2
votes
0answers
81 views

Fundamental Group of SL_2 [on hold]

I am thinking whether there is a simple criterion or visible method to know the fundamental group of SL_2(R), or SL_2(F) with an arbitrary field F. Because SL_2(R) is already a 3-dimensional ...
20
votes
9answers
1k views

Advanced Differential Geometry Textbook

I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help. In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...
1
vote
1answer
56 views

The Laplacian of an expression involving the Ricci tensor

While doing some computations on a compact Riemannian manifold I have reached the following expression: $$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$ where $\Delta_y$ is the ...
4
votes
1answer
130 views

Laplace-Beltrami and averaging

For a Riemannian manifold $M$ with metric $g$ and Laplace-Beltrami operator $-\Delta_{g}$, what conditions on $M$ guarantee that $-\Delta_{g} u(x)$ measures the difference between $u(x)$ and the ...
2
votes
0answers
53 views

Explanation that Twistor Space of $S^4$ is $\mathbb{C}P^3$?

I am attempting to read Atiyah's paper on self-duality in four-dimensional Riemannian geometry, and I came across the following basic example: Let $S_-$ be the $SU(2)$-bundle of anti-self dual ...
-1
votes
0answers
106 views

Is $(X_G, d_G)$ , compact manifold?

Let compact topological group $G$ acts on $(X,d)$ . We define a relation $\sim$ on $(X, d)$ as follows: for $x,y\in X$: $$x\sim y \Leftrightarrow x=gy \ \text{ for some } g\in G.$$ It is clear that ...
0
votes
1answer
193 views

Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me: Is there any open Ricci-flat ALE 4-manifold other than ...
2
votes
1answer
71 views

Set of regular points in an Alexandrov space with curvature bounded below

Let $X^n$ be an $n$-dimensional Alexandrov space with curvature bounded below. A point $x\in X$ is called regular if the space of directions $\Sigma_x$ is isometric to the standard sphere $S^{n-1}$. ...
7
votes
2answers
266 views

For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?

Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature? An obvious case is when $Y$ ...
1
vote
0answers
43 views

Doubling theorem for Alexandrov spaces

Is there a user friendly exposition of the notion of boundary of an Alexandrov space with curvature bounded from below and of the Doubling theorem? The only reference I am aware of is the original ...
5
votes
2answers
298 views

Riemannian metrics preserved by diffeomorphisms

Let $f \neq Id$ be a diffeomorphism (of a smooth manifold $M$) which admits some Riemannain metric on $M$ making it an isometry. How many different metrics are preserved by $f$? Note that ...
-1
votes
0answers
90 views

Curvature in geometry-interpretation

Previously this question was asked on stack exchange: the answer contained only reference to the wikipedia page which I already read (as mentioned in my post). So here is the question: The are ...
1
vote
0answers
60 views

Comparing Dirichlet energy and area of a Surface-immersion

Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...
5
votes
2answers
450 views

Square of the distance function on a Riemannian manifold

Let $(M^n,g)$ be a smooth Riemannian manifold. Consider the square of the distance function $$dist^2\colon M\times M\to \mathbb{R}$$ given by $(x,y)\mapsto dist^2(x,y)$. It is easy to see that this ...
8
votes
2answers
704 views

Does every manifold have a flat connection?

Suppose I have a manifold and a vector bundle over it, but not a connection or a metric. Can I always find a connection on it that has a Riemann curvature tensor that is identically zero? If so, can I ...
7
votes
3answers
591 views

Can a manifold have a curvature-free connection that is not torsion-free?

Suppose I have a smooth manifold with a tangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a ...
1
vote
0answers
95 views

When is a conformal class equal to a conformal orbit?

Let $(M,g)$ be a Riemannian manifold of dimension $n$. Let $\text{conf}(M,g)$ denote the conformal group, i.e. the subgroup of diffeomorphisms of $M$ that acts by conformal transformations relative to ...
8
votes
3answers
730 views

is there a global obstruction for a diffeomorphism to be an isometry?

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. We know ...
4
votes
2answers
248 views

Vector Fields in a Riemannian Manifold

Suppose $(M,g)$ is a Riemannian manifold. Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator? Thanks
2
votes
0answers
30 views

Generate matrices at the given distance from the initial matrix

I have n-by-n correlation matrix $C_{ij}$, it's symmetric positive definite, and $C_{ii}=1$ and $C_{ij}\le1$ etc. I want to generate a bunch of matrices $M_{ij}$ which are on the same distance from ...
4
votes
3answers
702 views

Jacobi fields on a “bump surface”

Consider a "bump surface" which looks like the following: Such a surface is rotationally symmetric, $C^2$-smooth, has positive curvature in the middle and negative curvature along the ring (the ...
1
vote
0answers
32 views

Normal fields of geodesic spheres

This question is related to this one (http://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres) I've asked at math.stackexchange. Let $(M,g)$ be a compact Riemannian ...
1
vote
2answers
237 views

Examples on small cut radius of totally convex set in non-negatively curved manifold

Suppose $M^n$ is an open complete nonnegatively curved Riemannian manifold. In Cheeger-Gromoll's proof of the soul theorem. They need an estimate on the cut radius of a totally convex set $C$. By a ...
3
votes
1answer
159 views

Orbits of Metrics under the Action of the Diffeomorphism Group

Consider the $n$-sphere $$ S^n = \{x\in\mathbb{R}^{n+1}: 1 - \sum_{k=1}^{n+1} x_k^2 = 0\}, $$ and let $g_1$ be the induced metric. Given $\lambda\in\mathbb{R}^{n+1}_{>0}$, we have the ellipsoid $$ ...
1
vote
1answer
72 views

The momentum constraints in the ADM formulation of general relativity

Suppose that the space-time has a time function. Let $g_{ij}$ be the Riemannian metrics of the time slices, and $K_{ij}$ be the second fundamental forms. It is by Codazzi equation that $$ ...
6
votes
1answer
320 views

Formula for the distance in noncommutative geometry

Probably the most famous formula in noncommutative geometry is the following formula allowing one to compute distance of two points using the operator theoretic data: $$(1) \ \ ...
2
votes
1answer
81 views

Limited expansion of mean curvature of geodesic spheres

I am working with the Laplacian on a Riemannian manifold $(M,g)$ (compact, without boundary). In spherical geodesic coordinates $(r, \sigma)$ around some arbitrary $x \in M$ (where $\sigma$ denotes ...
1
vote
0answers
52 views

Immersed surfaces in Hyperbolic 3-manifolds

Given a hyperbolic 3-Manifold $M=\Gamma_{0}\setminus\mathbb{H}^3$, and a smooth, connected, compact immersed negatively curved surface $\Sigma=\Gamma\setminus\widetilde\Sigma\subset M$, where ...
13
votes
3answers
579 views

Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators

EDIT: According to some comments on this post I revise the title to remove the misunderestanding. Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...
43
votes
2answers
2k views

“Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...
1
vote
0answers
102 views

understanding geometry of eigen values of Ricci tensor [closed]

As per I can visualize the eigen value $\lambda$ of a linear map $T:V \rightarrow V$, defined by $Tv=\lambda v$, is actually the scaling factor of the vector in the same direction as of $v$.My ...
1
vote
1answer
151 views

Symmetries of non-Riemannian curvature tensor

The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection. By construction it is antisymmetric in the first two indices, since roughly ...
0
votes
0answers
30 views

derivation of a expression in the ricci flow on surfaces

Recently I am studying benett chow and dan knopf's book titled Ricci flow:an introduction.In chapter 5 (Ricci flow on surfaces) I am stuck in a straightforward deduction.May be it is very simple,but ...
3
votes
1answer
133 views

Smooth manifolds for which every metric is geodesically convex

Are there non compact smooth manifolds which have the property that every Riemannian metric is geodesically convex? Note that a manifold for which every Riemannian metric is complete must be compact. ...
3
votes
1answer
97 views

Action generated by geodesic flow is hamiltonian

I'm trying to understand why a certain action of a Lie Group is hamiltonian. Let $(M,g)$ be a geodesically complete Riemannian manifold. Then there exists a canonical one-form on the cotangentbundle ...
3
votes
2answers
195 views

Triangles in rigid Riemann surfaces

Edit: We thank Vladimir Matveev for his comment on this post which leeds us to revise the question as follows: Assume that $M_{g}$ is a compact Riemann surface with constant negative cuvature (That ...
4
votes
2answers
296 views

What does it mean that the Hessian is proportional to the metric?

Let $(M,g)$ be a smooth manifold equipped with a metric tensor $g$, and $f\in C^\infty(M)$ a regular function (i.e., with nowhere vanishing differential). Denote by $\mathrm{Hess}_g(f):=\nabla df$ ...
10
votes
2answers
908 views

Formal adjoint of the covariant derivative

Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric. It is essentially ...
4
votes
0answers
85 views

Gromov's compactness theorem for manifolds with boundary

The Gromov's compactness theorem says that if $\{M_i^n\}$ is a sequence of closed Riemannian manifolds of dimension $n$ with uniformly bounded diameter and uniformly bounded from below Ricci curvature ...
17
votes
0answers
171 views

Metrics on the 3-sphere with knotted geodesics

According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic ...
3
votes
0answers
63 views

Barycentric interpolation in hyperbolic triangles

Let $T$ and $T'$ be triangles in the hyperbolic plane $\mathbb{H}^2$, denote by $A, B, C$ and$A', B', C'$ their vertices respectively. Let $f : T \to T'$ be the unique "barycentric interpolation" that ...
2
votes
0answers
74 views

$Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]

This is some sort of "follow-up" to the (unanswered) question posted here. Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$ Then $\varphi $ is an automorphism of $O(2n)$, and ...
4
votes
1answer
134 views

Non-flat totally geodesic surfaces

I'd like to know whether a Riemannian symmetric space of compact type admits a non-flat totally geodesic surface. I've found an article by Mashimo on the classification of these surfaces for certain ...
3
votes
0answers
45 views

Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]

Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the ...
2
votes
0answers
73 views

What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian?

Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$. How big can the set ...
0
votes
0answers
51 views

Toponogov comparison theorem for complex manifold

I want to know some reference for the Toponogov comparison theorem for complex manifold, in particular, for complex manifold with bounded holomorphic sectional curvature. As far as I know, the ...
0
votes
1answer
60 views

Sobolev chain rule on non-compact manifolds

Let $(M,g)$ be a non-compact Riemannian manifold (not of bounded geometry). Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$ with $f'$ bounded and $f(0)=0$. Is the Sobolev chain rule valid for functions ...
5
votes
3answers
607 views

Limit cycles as closed geodesics(geodesible flow)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$: \begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation} This equation defines a foliation on ...