All Questions
Tagged with dg.differential-geometry riemannian-geometry
1,985 questions
19
votes
2
answers
2k
views
Area of distance sphere in manifold with Ricci $\ge 0$.
Let $M$ be a open complete manifold with Ricci curvature $\ge 0$.
By a theorem of Calabi and Yau, the volume growth of $M$ is at least of linear.
I am wondering whether the following statement is true:...
- 891
17
votes
3
answers
2k
views
Why the quarter in the $\frac{1}{4}$-pinched sphere theorem?
Is there any hope of a high-level explanation of why the fraction $\frac{1}{4}$
plays such a prominent role as a
sectional curvature
bound in Riemannian geometry?
My (dim) understanding is that the ...
- 151k
5
votes
2
answers
917
views
Example for Busemann function is not an exhaustion when Ricci $\ge 0$
For an open complete Riemannian manifold $M$ with non-negative sectional curvature, the Busemann function defined below is a convex exhaustion function (by Cheeger-Gromoll's proof of soul theorem)
...
- 891
36
votes
10
answers
6k
views
Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature
A curve in the plane is determined, up to orientation-preserving
Euclidean
motions, by its curvature function, $\kappa(s)$.
Here is one of my favorite examples, from
Alfred Gray's book,
Modern ...
- 151k
5
votes
1
answer
927
views
Space of metrics with positive sectional curvature
Hello;
We know that the space of riemannian metrics on a compact manifold is an open cone in the space of symmetric 2-tensors.
Is it reasonable to think that metrics with positive sectional ...
- 469
4
votes
2
answers
1k
views
Invariant Metrics on the Sphere
I've been thinking about $SU(n)$-invariant metrics on the odd-dimensional spheres $S^{2n-1} \simeq SU(n)/SU(n-1)$. For $S^1$, all such metrics are in correspondence with the positive reals. For $S^{3} ...
- 3,399
8
votes
3
answers
1k
views
Higher derivatives than Jacobi fields
The first and second derivatives of the distance function (either the full $d:M\times M\to \mathbb{R}$ function or the $d(p,\cdot):M\to \mathbb{R}$ function) as well as the derivative of the ...
- 516
5
votes
2
answers
7k
views
Inner products on differential forms
Given a Riemannian metric $g$ on a smooth manifold $M$, one defines an
$L^2$-inner product on the space $\bigwedge^\ast(M)$ of differential
forms by
$$
\langle \alpha, \beta \rangle_g = \int_M \...
- 585
36
votes
3
answers
3k
views
When is a closed differential form harmonic relative to some metric?
Let $\omega$ be a closed non-exact differential $k$-form ($k \geq 1$) on a closed orientable manifold $M$.
Question: Is there always a Riemannian metric $g$ on $M$ such that $\omega$ is $g$-harmonic,...
- 585
4
votes
1
answer
995
views
Connection 1-forms of a Riemannian metric and the norm of the Hessian and ( seemingly ) two different definitions of Hessian and its norm
In the paper "On Quasiconformal Harmonic Maps " (link here) by L. F. Tam and T.Y.H. Wan, Pacific Journal of Mathematics, vol 182, no 2, 1998, in section 1, they define the Hessian of a function $f :H^...
- 1,471
7
votes
1
answer
1k
views
Helmholtz-Decomposition on compact Riemannian manifolds
For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty $ into a "gradient"- and a "divergence-free"-part such that
$L^...
- 73
4
votes
0
answers
873
views
Notion of distance for tangent vectors of a Riemannian manifold
I have the following question. Assume we have a Riemannian manifold $M$ with the induced metric given by $d$.
I am looking for a canonical way to compare two elements $v,w\in T^n M$, where $T$ ...
- 979
3
votes
2
answers
989
views
About Jacobi fields on nonpositive curvature
Let's work on a Riemannian manifold $M$ of nonpositive sectional curvature.
Fix a unit-speed geodesic $\beta$, and a Jacobi field $\eta$ over it. Assume that $\eta(0)$ is nonzero and orthogonal to $\...
- 2,479
4
votes
1
answer
723
views
Morse theory and adiabatic limits
Let's start with a product manifold $M\times N$, with a product Riemannian metric $g_M\oplus g_N$. Consider a generic Morse function $f(x, y)$ on $M\times N$. Then its critical points are discrete and ...
- 1,207
16
votes
0
answers
850
views
Applications of Berger's Curvature Estimate
I'm interested in applications of the following estimate of Berger on the Riemann curvature tensor:
Let $(M,g)$ be a Riemannian manifold of dimension $n \geq 4$, let $p \in M$, and assume that the ...
- 4,899
19
votes
2
answers
908
views
Manifold with all geodesics of Morse index zero but no negatively curved metric?
A closed oriented Riemannian manifold with negative sectional curvatures has the property that all its geodesics have Morse index zero.
Is there a known counterexample to the "converse": if (M,g) is ...
- 7,005
12
votes
2
answers
1k
views
Obstructions to Einstein metrics in high dimensions
It is well known that there exists three and four manifolds that do not admit an Einstein metric, but I wonder if this question is still open for manifolds of dimension higher than four. That is, does ...
- 1,701
10
votes
2
answers
1k
views
Einstein metrics and conformal geometry
I recall reading somewhere that if a conformal class contains an Einstein metric then that metric is the unique metric with constant scalar curvature in its conformal class, with the exception of the ...
- 1,701
11
votes
4
answers
3k
views
Laplace-Beltrami Operator on Surfaces
I would like to know what is known about the spectrum of the Laplace-Beltrami operator on 2-dimensional negatively curved surfaces of constant curvature.
For instance,
What is the spectrum of the ...
- 3,626
4
votes
0
answers
343
views
Constant scalar curvature+Constant $\sigma_2(C_g)$ curvature = ?
Let $(M,g)$ be a closed, smooth, Riemannian manifold of dimension $n>4$. Suppose both the scalar curvature and norm of the Ricci tensor are constant. In addition suppose that $g$ satisfies the ...
- 1,701
3
votes
0
answers
234
views
Negative Paneitz constant on $n$-sphere
Let $\Pi$ be the Riemannian functional defined on the space of Riemannian metrics on $S^n$, $n>4$, as follows:
$$
\Pi(g) = \int_M \frac{(n-4)(n^3-4n^2+16n-16)}{16(n-1)^2(n-2)^2} R_g^2 - \frac{2(n-4)...
- 1,701
27
votes
0
answers
3k
views
4
votes
0
answers
296
views
Minimality of geodesics on incomplete manifolds
On complete Riemannian manifolds, there is a characterization of the time $t_0$ when a geodesic $c$ stops being minimizing: either $c(t_0)$ is conjugate to $c(0)$ along $c$, or there exists a geodesic ...
- 306
6
votes
2
answers
903
views
Ricci curvature of the symplectic group
Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$?
For $O(n)$ and $U(n)$ I know many references which state such a ...
- 11.4k
8
votes
2
answers
2k
views
Riemannian manifolds that are scalar flat but not Ricci flat
What are the examples of Riemannian manifolds that have zero scalar curvature but non-zero Ricci curvature? Is there any sort of classification of such manifolds?
- 403
1
vote
0
answers
472
views
Finding a metric on a tubular neighborhood of an embedded surface
Hey all. The setup for my question is an embedded surface $\Sigma \hookrightarrow M$ in a smooth, compact 4-manifold $M$. Assuming one knows the induced metric $g_{\Sigma}$ on $\Sigma$, I would like ...
- 145
6
votes
1
answer
1k
views
Basic results in bounded geometry
I'm doing analysis (dynamical systems) in the context of Riemannian manifolds of bounded geometry and I find myself reproving quite a few standard results/tools from standard differential geometry, ...
- 2,481
8
votes
1
answer
2k
views
Calculating the geodesic equation for a particular set of phase-space coordinates
Let $g$ be a Riemannian metric on the $d$-dimensional flat space $\mathbb R^d$, and consider the usual Lagrangian $$L(x, \dot x) = \tfrac 1 2 g_{ij}(x) \dot x^i \dot x^j.$$ Let $\hat g := \sqrt g$ ...
- 8,512
11
votes
4
answers
4k
views
Eigenvalues of Laplacian-Beltrami operator
I am interested in the first non zero eigenvalue of the Laplace-Beltrami operator in a 2D compact manifold, and if there is a geometric characterization of its value.
I am interested in the case when ...
- 163
3
votes
0
answers
438
views
Is the Hessian almost everywhere nondegenerate?
Let $M$ be a complete Riemannian manifold. For a fixed point $p$ in $M$, the Riemannian distance to $p$ is denoted by $d_p$. Fix a strongly convex geodesic ball $B(o,R)$ in $M$ and some disjoint ...
- 265
0
votes
1
answer
218
views
Does an abelian group acting on a riemaniann manifold define an othogonal foliation?
This question is related to my previous question. Suppose that a group $G$ acts freely and properly on a Riemaniann manifold $(M, g)$. Than the orbits form a foliation for $M$. For $p \in M$, let $V_p$...
- 339
7
votes
5
answers
3k
views
"Famous" 2d Riemannian manifolds with non-constant curvature
I'm looking for "famous" or otherwise well-known 2d Riemannian manifolds which have non-constant curvatures but have a non-trivial Killing vector field. Of course there are tons of spaces like these, ...
- 362
10
votes
3
answers
862
views
Questions on smoothness of Riemann metrics
I've heard assertions of the sort:
Let there be a Riemann metric (not very smooth, say of class $C^1$ or $C^2$ or maybe $C$?) in a neighbourhood of a point on a manifold. Then it is possible to ...
- 2,715
4
votes
3
answers
2k
views
What is the defining formula for Sectional Curvature
What is the defining formula for sectional curvature?
$K_1(X,Y) = \frac{ \langle R(X,Y)Y, X \rangle} {\langle X,X \rangle \langle Y,Y \rangle - \langle X,Y \rangle} $
as in http://en.wikipedia.org/...
- 207
5
votes
3
answers
1k
views
equivariant index of Dirac Operator on $S^{2}$
First, I have to admit that I don't have much knowledge of Spin Geometry and Index Theory, the question could be too simple or naive and secondly there may be too many questions.
Let $D$ be the ...
- 3,218
6
votes
2
answers
2k
views
Geodesic completeness and complete Killing fields
Why are the Killing fields on a complete Riemannian manifold themselves complete (that is, the integral curves of the Killing fields are defined for all time)?
- 135
10
votes
2
answers
4k
views
Splitting of the double tangent bundle into vertical and horizontal parts, and defining partial derivatives
Let $M$ be a manifold and $g$ a metric on $M$.
Let $TM$ denote the tangent bundle of $M$, and denote points in $TM$ by $(x,v)$ where $v \in T_xM$.
The Levi-Civita connection of $(M,g)$ induces a ...
- 151
6
votes
1
answer
390
views
Do manifolds with no Ricci lower bounds for any metric exist?
Is there a smooth (noncompact) manifold $M$ such that for any Riemannian metric $g$ on $M$ there are $p_i$ and unit tangent vectors $v_i \in T_{p_i}M$ such that $Ricc(g)|_{p_i}(v_i,v_i) \leq -i$? This ...
- 7,197
8
votes
2
answers
2k
views
Kähler metrics for projective space that are not the Fubini-Study metric
For projective $N$-space $CP^{N}$, there is a canonical Kähler metric called the Fubini-Study metric. Do there exist other Kähler metrics for $CP^N$. If so, is there any classification of such metrics?...
- 1,479
2
votes
2
answers
763
views
Twisting Spinor Bundles with Line Bundles
In a paper I am reading, the following framework was given: Let $S$ be a spinor bundle, over a Riemannian manifold $M$, with Clifford action
$$
c:S \otimes \Omega^1(M) \to S.
$$
Moreover, let $E$ be ...
- 1,453
26
votes
3
answers
3k
views
Does Ricci flow depend continuously on the initial metric?
Consider a version of Ricci flow for which short time existence and uniqueness are known,
e.g. the Ricci flow on a closed manifold. Does the solution $g_t$ for small $t$ depend continuously on the ...
- 29.1k
12
votes
1
answer
896
views
Analytic Torsion in the Derived Category
I recently learned about analytic torsion and about the amazing Cheeger-Muller theorem identifying analytic and Reidemeister torsion for compact Riemannian manifolds.
Now analytic torsion is defined ...
- 23k
3
votes
2
answers
657
views
Reference for Almost-Kahler geometry
Is there any comprehensive reference for Almos-Kahler geometry or more generally to Almost- Hermitian geometry ?
- 1,236
15
votes
2
answers
1k
views
Is there a unified reason that there are an infinite number of geodesics between nonconjugate points on a compact manifold?
The proof of this statement seems to break into two really different arguments. So, I'm wondering if there is a better argument that can explain them both, or whether it's really just two theorems ...
- 6,069
4
votes
2
answers
1k
views
Special Killing Vector Fields
Consider $(M^{n},g)$ to be a Riemannian manifold and suppose that $X$ is a smooth non-trivial Killing vector field on $M$. Away from the zeros of $X$ we have a natural distribution $D$ of $(n-1)$-...
- 2,299
3
votes
2
answers
519
views
Submanifolds lying on the boundary of a convex domain
Let $M$ be a submanifold of $\mathbb R^n$. Call $M$ locally convex if locally $M$ is contained in the boundary of a convex domain of $\mathbb R^n$.
Is there any known condition that is equivalent to ...
- 971
3
votes
1
answer
426
views
Restriction of the Levi-Civita Connection to a Connection on the (Anti-)Holomorphic Forms
For a Riemannian manifold $(M,g)$, that is also a complex manifold, when does the Levi-Civita $\nabla_g$ connection restrict to a connection on the holomorphic forms $\Omega^{(\cdot,0)}$, and when ...
- 1,462
5
votes
3
answers
5k
views
Dual Riemannian metric and the Dual Metric Form
Let $M$ be a Rieamnnian manifold with metric $g: X(M) \times X(M) \to C^{\infty}(X)$, where $X(M)$ are the vector fields of $X$.
As is well known, we can induce a bilinear pairing
$$
\langle \cdot , ...
- 1,462
3
votes
3
answers
2k
views
Gaussian curvature radius
In the paper Surface sampling and the intrinsic Voronoi diagram (2008), Ramsay Dyer defines the Gaussian curvature radius at a point $x$ of a surface $S$ to be $\rho_K(x) = 1/\sqrt{K(x)}$ where $K(x)=\...
- 579
8
votes
1
answer
916
views
geodesic 2-dimensional submanifolds of a Riemannian manifold [duplicate]
Possible Duplicate:
Must a surface obtained by exponentiating a plane in a tangent space of a Riemannian manifold be geodesically convex?
The one dimensional geodesic submanifolds of a given ...
- 11.1k