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.

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217 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 ...
2
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0answers
65 views

Approximating solutions to minima of the discrete Lagrangian

I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious. General gist of the problem I have a variational problem on a ...
2
votes
1answer
202 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 ...
4
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1answer
132 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 ...
8
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0answers
289 views

Reference - Asymptotic geodesics on compact surfaces without conjugate points

I would like to ask about possible references on the following problem: consider a compact surface and a metric without conjugate points. Consider it's universal covering endowed whith the lifting of ...
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1answer
430 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 ...
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0answers
137 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 ...
2
votes
1answer
257 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 ...
3
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1answer
317 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?
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0answers
160 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 ...
4
votes
1answer
167 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 ...
17
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2answers
711 views

Is every topological (resp. Lie-) group the isometrygroup of a metric space (resp. Riemannian manifold)?

The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers) So, is every topological ...
7
votes
2answers
331 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 ...
10
votes
3answers
568 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 $B\in Sym^2(M)$ are ...
0
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1answer
265 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 ...
17
votes
4answers
1k 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 ...
5
votes
2answers
322 views

Compact surface with genus$\geq 2$ with Killing field

Let M be a compact Riemannian surface of genus$\geq 2$. Can M have a globally defined Killing field ? Can M have a Killing field defined on M-(finite set of points)?
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0answers
158 views

Tangent space, metrics etc. on simplicial sets

Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting? ...
2
votes
2answers
411 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)
2
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1answer
195 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 ...
10
votes
1answer
343 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. ...
6
votes
1answer
425 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... ...
3
votes
1answer
246 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 ...
3
votes
0answers
150 views

Is there a way to metricize the notion of $C^\infty$ convergence of pointed Riemannian manifolds?

A sequence of pointed Riemannian manifolds $(M_n,p_n,g_n)$ is said to converge $C^\infty$ to pointed Riemannian manifold $(M,p,g)$ if for each positive radius $R$ there exists sequence of embeddings ...
7
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1answer
230 views

Smoothing of piecewise Euclidean Riemannian metrics

Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on ...
4
votes
2answers
374 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 ...
8
votes
6answers
2k 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 ...
13
votes
1answer
339 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
votes
2answers
397 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 ...
4
votes
0answers
214 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 : ...
6
votes
1answer
200 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 ...
4
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0answers
145 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 ...
1
vote
2answers
220 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 ...
5
votes
3answers
883 views

Totally Geodesic Submanifolds

Suppose that $N$ is a totally geodesic submanifold of a complete Riemannian manifold $(M,g)$. Is it the case that a geodesic segment that minimizes length in the submanifold $N$ also minimizes length ...
11
votes
1answer
356 views

Is a manifold with flat ends of bounded geometry?

A Riemannian manifold $(M,g)$ is said to have flat ends if the curvature tensor of $g$ vanishes outside a compact set $K$. I was wondering if such manifolds are of bounded geometry. Recall that a ...
3
votes
2answers
290 views

Uniqueness of Kähler form with same volume

Hallo, Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure ...
3
votes
0answers
175 views

What is known about analogous results of Kazdan and Warner in higher dimensions?

First let me state a Theorem due to Kazdan and Warner: ``Let M be a compact two dimensional orientable manifold. Let $f: M \rightarrow \mathbb{R}$ be a function that has the same sign as ...
8
votes
3answers
357 views

twisted Poisson structures, degenerate metrics and integrability properties of (2,0)-tensors

Given a regular (constant rank) bi-vector $\Pi \in \Gamma(\bigwedge^2TM)$ on a smooth manifold $M$ the necessary and sufficient condition for the image of $\Pi^\sharp:T^*M\to TM$ to be an integrable ...
3
votes
1answer
350 views

Grassmannian of oriented real $k$-planes

The Grassmann manifold $\widetilde{Gr}(k,\Bbb{R}^n)$ of oriented $k$-planes in $\Bbb{R}^n$ is a double cover of the Grassmann manifold $Gr(k,\Bbb{R}^n)$ of non-oriented $k$-planes. We can give ...
4
votes
1answer
315 views

Rigorous solution to Ricci Flow on dumbbell $S^3$

To begin a small interest in Ricci Flow and similar tools, I am starting with Hamilton's expository paper The Formation of Singularities in the Ricci Flow. This was posted in 1995, so I am wondering ...
6
votes
1answer
195 views

Hamiltonian polar action with Lagrangian section

I am looking for examples of Hamiltonian polar isometric actions of a compact Lie group on a Kahler-Einstein (or perhaps just Kahler) manifold, that admits a Lagrangian section. Recall that an ...
8
votes
1answer
263 views

Discretization of a complete manifold

Suppose $M$ is a complete Riemannian manifold with very large injectivity radius (say larger than $100$) and $\left\lbrace x_i: i \in I\right\rbrace$ is a maximal $1$-separated subset of $M$. Is ...
1
vote
1answer
140 views

Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold

Hallo, Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler ...
6
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0answers
185 views

Different complexifications of a real analytic Riemannian manifold

Hi, I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well knwon fact that in a neighbourhood $U$ of the ...
3
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3answers
301 views

Is the set of all smoothed closed simple curves on $\mathbb{R}^2$ a manifold?

In the studies of active contours they describe the set of all simple smooth closed curves on $\mathbb{R}^2$ to be a Riemannian Manifold $M$. The tangent space at a curve $c$, $T_cM$ is a set of ...
3
votes
2answers
237 views

Real analytic submanifolds of $\mathbb{R}^{n}$

Hallo, Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as ...
2
votes
1answer
189 views

Isometric embedding of a compact Lie Group in $M(n,\mathbb{C})$

Greetings, Let $G$ be a compact Lie group with a bi-invariant inner product $h$ on it. Can one embedd $G$ in $M(n,\mathbb{C})$ isometrically for some $n \in \mathbb{N}$. By isometrically I mean that ...
5
votes
1answer
428 views

What are the Dirac operators on $S^1$?

This is crossposted at stack exchange as http://math.stackexchange.com/questions/248391/dirac-operators-on-s1. I am trying to understand the Dirac operators associated to the 2 spinor bundles on ...
1
vote
1answer
209 views

Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold

Hallo, It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught ...
8
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0answers
347 views

“Homogeneity” of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]

My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...