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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Exhaustion of an open manifold of bounded curvature and finite volume

In the Cheeger-Gromov paper "On the Characteristic Numbers of Complete Manifolds of Bounded Curvature and Finite Volume", http://www.maths.ed.ac.uk/~aar/papers/cheegergr1.pdf, the authors make the ...
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Degeneration of riemannian metrics with curvature bounds

In short, I'm curious to know what modes of degeneration of metric might still keep the curvature bounded. More precisely, assume we are keeping the total volume of the manifold fixed and deform the ...
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188 views

Star-shaped domain in a space form

Let $M$ be either $\mathbb R^n$, $\mathbb H^n$ or $\mathbb S^n$ and $p\in M$, by a star-shaped domain w.r.t $p$ I mean a connected open subset $\Omega$ in $M$ containing $p$ such that its boundary is ...
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Are negatively pinched manifold locally conformally flat?

One knows that hyperbolic manifolds are locally conformally flat. How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy: $$ -\Lambda \le K \le -\lambda$$ for ...
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1answer
277 views

Collapsing of Riemannian manifolds with a group action

Let $M$ be a complete Riemannian manifold with bounded sectional curvature and $G$ a compact connected Lie group acts smoothly on $M$. Consider the fixed point set $F$, it is of course a submanifold ...
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$J$-holomorphic curve as a minimal surface

The following is a part of the proof of Gromov nonsqueezing theorem. The existence of a $J$-holomorphic curve gives an upper bound for the radius of a symplectically embedded ball. Let $\psi: B(r) ...
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Almost constant bump function

I ran into the following situation and it turned out to be more subtle than it looked. I have a complete Riemannian manifold $M$ and my objective is to construct a sequence of functions $f:M \to ...
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289 views

Complete metric on a Riemann surface with punctures

If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric? I know that in this case the universal cover is the hyperbolic plane ...
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290 views

Volume of a Riemannian manifold and its relation to fundamental group

I am reading a book (Mapping Class Group by Farb and Margalit) and it says (in a proof of one theorem): If $S$ admits a hyperbolic metric (they define such a surface to be of finite area and ...
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348 views

Nearly constant curvature implies “nearly isometric” to a space form?

It is well known a Riemannian manifold with constant sectional curvature is a quotient of the Euclidean space, hyperbolic space or sphere. In particular we know how their metric looks like locally. ...
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1answer
195 views

Sobolev Norm of distance function on Riemannian manifold

Suppose $M$ is a Riemannian manifold with distance function $d:M\times M \rightarrow [0,\infty)$. If it helps let $M$ be a Lie group with finite Haar measure $\mu$ and left invariant metric (like ...
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1answer
271 views

Extension of groups in Bieberbach's theorem

I am reading de la Harpe's book "Topics in Geometric Group Theory". On page 145, there is a theorem: Let $V$ be a complete $n$-Riemannian manifold with sectional curvature satisfying $K\ge 0$. Then ...
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248 views

holomorphic extension of forms

hallo, I have the following question: Let $M$ be a $n-$dimensional complex manifold and $X \subset M$ be a compact $n-$dimensional totally real analytic Riemannian submanifold. Let furthermore ...
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1answer
137 views

estimate over simply-connected Riemannian manifold with non-positive sectional curvature

Let$M$ be a Complete simply-connected n-dimensional Riemannian manifold with nonpositive curvature,$\Omega $is a open subset of $M$ ...
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220 views

Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold?

Let $M^{n-1}$ be a closed flat manifold. Is it true that there exists a hyperbolic manifold $N^n$ with finite volume such that $M$ is a cusp cross-section of $N$? It was proved in "On the geometric ...
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1answer
284 views

Minimal distance spheres in complex projective spaces

My question has to do with distance spheres in $\mathbb CP^{n+1}$. I am interested in knowing what is the radius $r$ of a distance sphere $S(r)$ around a point that makes it a minimal submanifold ...
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1answer
589 views

Totally geodesic submanifold of round sphere

Let $S^n$ be the $n$-dimensional round sphere (i.e. with Riemannian metric of constant curvature +1). Is there any classification result of totally geodesic embedded submanifolds? Are they all round ...
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Negative Einstein manifolds

In Besse's "EInstein manifolds", p. 354, the question is posed if the volume of Einstein metrics on a given compact manifold (normalized such that $Ric=\pm(n-1)g$) take only finitely many values. For ...
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1answer
238 views

G-structures and complete riemannian manifolds

what are possible fundamental and introductory texts about G-structures ? and where i can find the proof of this proposition: if G(group) acts properly discontinuously on a space X , then G is a ...
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447 views

Parallel translation on surfaces

Parallel translation of a vector along a geodesic in a surface is characterized by the following three properties: The vector being transported moves continuously. It has constant norm. It maintains ...
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1answer
164 views

cone angle at infinity for product of cones

Let $A=\lim_{r \rightarrow +\infty} \frac{Vol(B(o,r))}{\omega_{n} r^{n}}$ for any Riemannian manifold $(\mathbb{M}^{n},g)$ with nonnegative Ricci curvature. Here $\omega_{n}$ is the volume of unit ...
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1answer
463 views

Isoperimetry and Poincare Inequality

What are the known relations between isoperimetric and Poincare inequalities on manifolds? For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
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1answer
524 views

A strange question about closed geodesics on a closed manifold

I'm studying a particular kind of curve evolution on Riemannian manifolds. It would help me to know the answer to the following kinda weird question: Does there exist a closed Riemannian manifold $M$ ...
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177 views

Understanding Paul Lévy Theorem [Riemannian Geometry]

In Gromov's paper regarding this proof, he says that by taking the best possible $H$ , we can get the following result: Let $k>0 , x$ be such that $ 1-x \leq K \int_0^{\infty} J_{k,H} (t) dt $ and ...
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1answer
412 views

Heisenberg group: research themes

I am currently studying the Heisenberg group from the Riemannian geometry point of view, particularly focusing on its Gromov boundary and more generally its metric properties. I would like to know ...
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362 views

How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?

To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$. Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$. Prove: Suppose $f\in C^\alpha(M)$ satisfies ...
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Manifold-Valued Sobolev Spaces

I have the following basic question about Sobolev-spaces which take their values in a Riemannian manifold $(M,g)$, i.e. functions $u:\Omega \to M$, $\Omega \subset \mathbb{R}^n$ bounded, such that ...
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547 views

Kähler potentials that depend only on geodesic distance

Hermitian symmetric spaces of constant curvature have the property that the potential for their Kähler metric can be expresed as some function of the geodesic distance. Does anyone know if there are ...
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486 views

Short time existence on Hyperbolic Ricci flow in non-compact case

We know Laplace equation (elliptic equations) $ Δ u = 0$ Heat equation (parabolic equations) $u_t − Δu = 0$ Wave equation (hyperbolic equations) $u_{tt} − Δu = 0$ we have - Hyperbolic geometric ...
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1answer
207 views

estimate of metric tensors in terms of curvatures

I would appreciate if someone knows how to get the following estimates: Let $\rho_m$ is a sequence of real numbers approaching $\infty$. Consider a sequence of Riemannian metrics $g^{(m)}$ on $S^3$ ...
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0answers
132 views

Deforming isometric embeddings in low codimension

Let $F:M\to \mathbb R^N$ be an embedding. This embedding induces a metric $g_F=dF\cdot dF$ on $M$, that turns $F$ into an isometric embedding. Probably the hardest part of the proof of the Nash ...
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1answer
401 views

Geometric conditions for isoperimetric, Sobolev, Poincar\'e inequalities on a riemannian manifold

By a theorem of Lichnerowicz, on a riemannian manifold $M^{(m)}$ with positive Ricci curvature, the reciprocal of Sobolev constant(ie. the first eigenvalue of laplacian) can be bounded from below by ...
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624 views

Constant scalar curvature metrics in a conformal class

Let $(M,g)$ be a compact Riemannian manifold, then by the resolved Yamabe-problem, there exists a metric $\tilde{g}$ of constant scalar curvature in the conformal class $[g]$ of $g$. By normalizing ...
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Does the curvature determine the metric?

Hello, I ask myself, whether the curvature determines the metric. Concretely: Given a compact Riemannian manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that ...
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1answer
223 views

Constant Mean Curvature hypersurfaces “condensing” onto a minimal submanifold

Let $M$ be Riemannian manifold and $S\subset M$ a minimal submanifold, with $\dim S<\dim M-1$. According to a few references (e.g., Mahmoudi, Mazzeo & Pacard), it should not be hard to see ...
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1answer
601 views

recognizing Kahler manifolds of complex dimension n

Is there new classification of Kahler manifolds of complex dimension n and new results for necessary and sufficient conditions for a manifold being Kahler? I know if redactivity of Lie algebra on ...
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635 views

Tweetable way to see that Willmore energy is Möbius invariant?

Consider a compact orientable Riemannian manifold $M$ (without boundary) isometrically immersed into $\mathbb{R}^3$. The Willmore energy of $M$ is the functional $$\mathcal{W} = \int_M H^2 dA$$ ...
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1answer
315 views

embedding torus [closed]

could anyone please help me? why is it impossible to embed a torus in R^3 with index 1 ( usual euclidean space with index 1 as a semi-riemannian manifold) as a semi-riemannian submanifold? thanx. ...
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1answer
325 views

Preissmann and Byers Theorems

I'm starting to study at the elementary level the relationship between topology and geometry of a Riemannian manifold of negative curvature. The first two theorems, simple and interesting in this ...
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1answer
167 views

relation with jacobifields in a small neighbourhood

hi, I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...
3
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1answer
373 views

Geodesic circles on riemannian manifolds

Can one always find, in a compact riemannian manifold, a closed geodesic isometric to a usual circle when endowed with the ambient distance ? For instance, in the usual flat torus, the only geodesics ...
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2answers
373 views

Metric Deformations from Non-Negative to Positive Curvature

Is it possible to deform the metric $g$ of a closed Riemannian manifold $(M,g)$ satisfying $\mathrm{Ricci}(M,g) > 0$ and $\mathrm{sec}(M,g) \geq 0$ to a metric $g_1$ satisfying $\mathrm{sec}(M,g_1) ...
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1answer
525 views

Conformal Killing spinors

In general I would like to know about the significance of conformal Killing spinors (especially keeping in mind supersymmetric theories on curved space-time). If $\epsilon$ and the $\bar{\epsilon}$ ...
4
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2answers
413 views

Is the exponential map of a $C^{1,1}$ Riemannian metric a local homeomorphism?

Suppose that $g$ is a $C^{1,1}$ (i.e., continuously differentiable with locally Lipschitz first derivative) Riemannian metric on a smooth manifold $M$. It seems to be known that locally the ...
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1answer
431 views

Are isospectral manifolds necessarily homeomorphic?

It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric. Is it known if there are closed Riemannian manifolds which are isospectral but not ...
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523 views

Adjoint of a Connection Using the Hodge Map?

For a Riemannian manifold $(M,g)$ with exterior derivative d, the codifferential d$^\ast$ is defined to be the unique map for which $$ g(\omega,d\omega') = g(d^* \omega,\omega'), ~~~ \omega,\omega' ...
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551 views

Levy-Gromov Isoperimetric Inequality

In his paper "Paul Levy's Isoperimetric Inequality", Gromov gives the following isoperimetric inequality: Let $V$ be a closed $(n+1)$-dimensional Riemannian Manifold with $\mathrm{Ric}(V) \geq n ...
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419 views

the left hand side of the Ricci flow equation at the initial value

I just started to learn about the Ricci flow and try to understand the Ricci flow evolution equation. It states that a one-parameter family $g_t$, $t\in[0,T)$ of Riemannian metrics on a smooth closed ...
3
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1answer
275 views

Identity of the Weyl-Tensor

Let $(M^n,g)$ be a Riemannian manifold and let $W$ be its Weyl tensor. For a given ONB, does the identity $$W_{ijkl}W_{ijkm}=\frac{1}{n}|W|^2g_{lm}$$ hold? I think I've seen it somewhere but I'm not ...
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3answers
442 views

Positively curved metrics on $S^2\times S^2$

As you know, the Hopf conjecture is about the existence of positively curved metric on $S^2\times S^2$. Hsiang-Kleiner have shown that there exists no positively curved metric admitting $S^1$-action ...