All Questions
Tagged with dg.differential-geometry riemannian-geometry
1,985 questions
9
votes
1
answer
539
views
Geometry defined by foliation.
In $\mathbb R^3$ there are 3 natural foliations given by the lines parallel to each axis, which intersect transversally. Let $M^n$ a manifold with $n$ foliations by lines or circles that intersect ...
7
votes
0
answers
996
views
On Perelman's paper
In section 5 in "The entropy formula for the Ricci flow and its geometric applications" Perelman has written:
Fix a closed manifold $M$ with a probability measure $m$, and suppose
that our system is ...
- 1,303
5
votes
2
answers
2k
views
Triangle area on surfaces of constant curvature
I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length.
Of course, the ...
- 9,853
4
votes
2
answers
483
views
Positively curved manifold with a codimension 1 totally geodesic submanifold.
Fact : Consider the inclusion $V^{n-1} \rightarrow M^n$ where $M$ is a closed orientable simply
connected positively curved manifold.
Then connectivity lemma implies that the inclusion is $(n-1)$-...
- 1,100
2
votes
1
answer
961
views
Ricci flow as a gradient flow and its Lyapunov function
In study of Ricci flow, for making Ricci flow as a gradient flow I faced $\mathcal{F}(g,f)=\int (R+|\nabla f|^2)e^{-f}$. I know that if we suppose $\frac{df}{dt}=-R$, then $\frac{d}{dt}\mathcal{F}(g,f)...
- 1,303
2
votes
2
answers
442
views
Holonomy groups of quotient Riemannian manifolds?
Let $(X,g)$ be a Riemannian manifold with holonomy group $Hol(X,g)$. Suppose that a finite group $G$ acts on $X$ freely and the metric $g$ is invariant under $G$. What can one say about the the ...
- 21
9
votes
1
answer
342
views
Positively curved manifold with almost extreme diameter
Suppose $M$ is a 1-connected closed manifold with sectional curvature $\ge 1$. So the diameter $D$ of $M$ satisfies
$$
D \le \pi
$$
When equality holds $M$ is isometric to round sphere. In fact this ...
- 2,623
6
votes
2
answers
892
views
Riemannian manifolds with small geodesics and bounded curvature
Let $(M,g)$ be a compact riemannian manifold with sectional curvature $|K_g| \leq 1$. A lemma due to Klingenberg asserts that then either the injectivity radius $i_g \geq \pi$ or $(M,g)$ contains a ...
- 847
3
votes
1
answer
301
views
Is geodesic plane field a Killing field?
Let $M$ be a closed orientable Riemannian manifold. Recall that a plane field on a Riemannian manifold is said to be geodesic if any geodesic tangent to the plane field at one point is tangent to it ...
- 283
4
votes
2
answers
2k
views
Energy functional
During my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works $\mathcal{F}(g,f)=\...
- 1,303
1
vote
1
answer
292
views
Homogeneous Spaces and Equivariant Hodge Maps
For a homogeneous space $G/H$, endowed with a $H$-equivariant metric $g$, let $\ast$ be the corresponding Hodge star map. It seems that $\ast$ must also be $\ast$-equivariant, but I can't see how one ...
- 637
4
votes
1
answer
725
views
On Dimension of Instanton Moduli Space
I am reading Charles Nash's book on differential topology and QFT. In particular, I have question on the part calculating dimension of instanton moduli space. The question split into conceptual part ...
- 857
5
votes
1
answer
2k
views
Prescribing the Lie derivative of the metric?
This is a question that arises from my research problem. Suppose $(M,g)$ is a compact Riemannian manifold with boundary and $g$ is smooth up to the boundary (if you like, take $M$ to be diffeomorphic ...
- 51
34
votes
7
answers
16k
views
geometric interpretation of Lie bracket
On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written:
We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to
which the integral ...
- 1,303
4
votes
1
answer
467
views
Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold
Hallo,
I have two questions where I do not really know how to deal with them. Let $(M,J,g)$ be a Kähler manifold, where $g$ is the Riemannian metric and denote by $\omega(\cdot , \cdot) = g(J \cdot ,...
- 53
2
votes
2
answers
356
views
Geometry of Hopf fibrations and the fibration of Steifel Manfiolds over Grassmannians
When $F = \mathbb{R}, \mathbb{C}$ or $\mathbb{H}$, there are fibrations $$O(k,F)\rightarrow V_k(F^n)\rightarrow G_k(F^n)$$ where $V_k(F^n)$ are Steifel manifolds and $G_k(F^n)$ are Grassmannians. When ...
2
votes
1
answer
478
views
Is there a lower bound for variance in terms of curvature?
If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero (...
- 16.6k
7
votes
1
answer
1k
views
About Sectional Curvature [closed]
In a paper by Yann Ollivier:
Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint
of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...
- 1,303
2
votes
1
answer
331
views
What kinds of manifolds admit concave boundary?
We can find many examples of smooth Riemannian manifolds with boundaries whose boundaries are convex. But it seems to me I know no any example of smooth Riemannian manifold with concave boundary. So ...
- 359
3
votes
1
answer
174
views
A k-form is thought of as measuring the flux through an infinitesimal k-parallelepiped
On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal $k$-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of ...
- 1,303
3
votes
1
answer
583
views
geometric meaning of Ricci-flatness
What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, ...
- 1,303
2
votes
1
answer
260
views
Different notions of geodesics.
Let $M$ be a (without boundary and not necessarly complete) Riemannian manifold.
A map $c\colon [a,b]\rightarrow M$ is called geodesic of type A iff $c$ is piecewise smooth, parametrized proportional ...
- 29
24
votes
7
answers
5k
views
Difference between parallel transport and derivative of the exponential map
This is a crosspost from math.stackexchange
Given a Riemannian manifold $M$, let $c(t) = \exp_p(tX)$ be the geodesic emanating from $p \in M$ with initial value $X$. Let $t_0$ be small enough, then ...
- 9,853
6
votes
1
answer
1k
views
Is it true that the geodesics on SO(n) and SU(n) are closed?
I mean for the bi-invariant metric (but actually any metric would work). In this metric geodesics are translates of 1-parameter subgroups so we need only to show that $exp(t X)$ for any X in the lie ...
- 803
3
votes
1
answer
438
views
positive sectional curvature of submanifold in $R^n$?
Let $N$ be a hypersurface in $\mathbb R^n$, assume it is compact. Then the maximum point of $d(O, x)$ when restrict to $N$ has positive sectional curvature lower bound by the one of the correspond ...
- 93
3
votes
1
answer
523
views
Diameter estimate of distance sphere of positive curved manifold
Let $M$ be an $n$-dimensional Riemannian manifold with sectional curvature lower bound 1. Fix a point say $O\in M$, let $S(r)$ denote the distance sphere centered at $O$ with radius $r$. The classical ...
- 2,623
4
votes
1
answer
208
views
Special coordinates for periodic metrics
This question is a follow-up to that one.
Given a $\mathbb{Z}^n$-periodic metric $g$ on $\mathbb{R}^n$ (with $n>2$), is it possible to find a periodic diffeomorphism $\varphi$ such that $\varphi^*...
- 14.4k
27
votes
1
answer
2k
views
Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood?
Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to ...
- 25.3k
1
vote
0
answers
175
views
The shape operator and an almost contact structure of a real hypersurface in $\mathbb{C}^n$
Let $S$ be an immersed real hypersurface in the Euclidean $\mathbb{C}^n$ with the standard complex structure $J$. Let $A:T(S)\rightarrow T(S)$ be the shape operator of $S$ (e.g. w.r.t. the outer ...
- 93
3
votes
1
answer
809
views
Induced Riemannian metric on Jet-Manifold
Suppose $(M,g)$ and $(N,g')$ are smooth Riemannian manifolds and $J^r(M,N)$ is the
smooth manifold of $r$-jets $j^r_xf$ of smooth maps $f:M\to N$.
Is there an 'induced' Riemannian metric $g''$ on $J^...
- 624
3
votes
1
answer
918
views
$C^k$ topology of metrics
Is the space of Riemannian metrics, over a compact manifold, complete when endowed with the $C^k$-topology of metrics?.
Is there a good reference for this?
- 123
0
votes
0
answers
752
views
Hessian of the inverse exponential map on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold. Then define
$f: T^*M \times M \to \mathbb{R}$
$f(x,\xi, y) = \langle exp_x^{-1} y, \xi \rangle$
where $exp_{\cdot}\cdot$ is the the exponential map and it's ...
- 225
4
votes
1
answer
305
views
Does convex set in Alexandrov space has positive reach?
Let $M$ be a metric space, $A$ a subset of $M$. The reach (defined by Federer) of $A$ in $M$ is the largest $r_0\ge 0$ such that if $x\in M$ and the $d(x, A)< r_0$, then $A$ contains a unique point ...
- 283
8
votes
2
answers
563
views
Is displacement controled by stable norm?
Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a $\...
- 14.4k
15
votes
3
answers
2k
views
Characterizing Hessians among symmetric bilinear tensors
I apologize in advance if this is somewhat elementary, but:
Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ ...
- 5,891
1
vote
1
answer
994
views
Lie derivative of curvature
Let $M$ be a Kahler manifold, with Kahler metric $g$. Let $X$ be a holomorphic Killing vector field of $g$, i.e. $L_{X} g = 0$, where $L_{X}$ is the Lie derivative along $X$. Let $R$ be the Riemannian ...
- 248
23
votes
4
answers
5k
views
Why is it important that partial derivatives commute?
I am asking this in the context of differential geometry (specifically Riemannian).
When the Levi-Civita Connection is defined, we require that the torsion tensor is 0, which in local coordinates ...
- 995
6
votes
2
answers
3k
views
When a Riemannian manifold is of Hessian Typ
When a Riemannian manifold is of Hessian Type (i.e., a Riemannian manifold which its metric is Hessian)
user21574
3
votes
1
answer
1k
views
Positivity of second fundamental form implies global convexity?
Let $M$ be a Riemannian manifold of dimension $n$. Let $N\subset M$ be a subset with smooth boundary $\Sigma=\partial N$. If one assume the second fundamental form $II$ with respect to inner normal ...
- 283
14
votes
1
answer
1k
views
Scalar curvature notion for Cartan connections
In Riemannian geometry, there is a well-known notion of the scalar curvature on a Riemannian manifold $M$, which is a function on $M$ given by a suitable contraction the Riemannian curvature tensor. ...
- 2,399
9
votes
1
answer
1k
views
How submanifolds evolve under Ricci flow?
This may be very naive, since I just started trying to learn Ricci flow; but I couldn't really find any answer after looking for a while in all the textbooks and lecture notes I found online...
...
- 5,891
8
votes
2
answers
2k
views
Applications of Gauss-Bonnet theorem
In wikipedia,I was pretty amazed to find a proof of fundamental theorem of algebra
using Gauss Bonnet theorem.
I think given how central it is to mathematics with its far reaching generalizations ...
3
votes
1
answer
410
views
First eigenvalue of $\Delta$ on Kaehler manifold with $Ricci\ge k$.
Let $M$ be a Kaehler manifold of complex dimension $n$. Let $\Delta$ be the real Laplacian of the underline Riemannian manifold. Let's assume the Ricci curvature of $M$ satisfies $\text {Ric}\ge k>...
- 283
6
votes
3
answers
2k
views
What is the Weitzenböck formula for the $\bar\partial$-Laplacian
It is well-known that the Weitzenböck formula for the real Laplacian is
$$\frac12 Δ|∇f|2=|Hessf|2+⟨∇f,∇Δf⟩+Ricci(∇f,∇f)$$
where $Hess$ denotes the Hessian tensor of $f$. and $\nabla f$ denotes the ...
- 93
19
votes
6
answers
9k
views
Tensor contraction and Covariant Derivative
What is the importance and intuition behind the the contraction operator on tensors (or the trace of a matrix, for that matter)?
In addition, I see that one of the requirements for a covariant ...
- 995
14
votes
1
answer
738
views
Algebraic characterization of the curvature operator of symmetric spaces
My question is the following :
Given an algebraic curvature operator $R\in S^2_B(\Lambda^2\mathbb{R}^n)$, is there an a simple criterion to know if this curvature operator can occur as the ...
- 4,123
7
votes
2
answers
2k
views
Cutlocus and conjugate points
I am thinking about the following questions about the cutlocus of a point in a Riemannian manifold or of a hypersurface in the Euclidean space:
1) If all the points of the (nonvoid) cutlocus of a ...
- 1,309
6
votes
0
answers
352
views
How to generate a random (Weyl) curvature operator ?
Given a dimension $n$, the space of curvature operators is the space $S^2_B(\Lambda^2\mathbb{R}^n)$ of symmetric endomorphisms $R$ of $\Lambda^2\mathbb{R}^n$ which satisfy the first Bianchi identity :
...
- 4,123
6
votes
1
answer
306
views
Fattening of totally convex sets
Suppose $(M, g)$ is an open complete nonnegatively curved Riemannian manifold with $d$ its distance.
A totally convex set $C\subset M$ has the property that for any two point $x, y \in C$ any ...
- 1,309
6
votes
0
answers
260
views
Can a simple Riemannian metric on the disc be extended to a Zoll metric on the sphere?
Given a simple Riemannian metric $(D,g)$ on the two-disc---its geodesics have no conjugate point and the boundary of the disc is strictly convex---, is it possible to embed $(D,g)$ isometrically into ...
- 13.5k