Questions tagged [riemannian-geometry]
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.
3,082 questions
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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. ...
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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 ...
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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>...
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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 $...
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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 ...
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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 ...
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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 $...
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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 :
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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 ...
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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 ...
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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 $\chi(M)$,...
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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 ...
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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 ...
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What are the Dirac operators on $S^1$?
This is crossposted at stack exchange as https://math.stackexchange.com/questions/248391/dirac-operators-on-s1.
I am trying to understand the Dirac operators associated to the 2 spinor bundles on $S^...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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"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 ...
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Are there countably many diffeomorphism classes of finite radius balls of complete Riemannian manifolds?
Suppose $M$ is a smooth complete Riemannian manifold and $x$ is a point in $M$. For any positive radius $r$ we consider the open ball $B(x,r)$ centered at $x$ with radius $r$.
If we ignore the ...
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What are first eigenfunctions of Laplacian for $CP^n$ with Fubini-Study metric?
I know the round $n$-sphere has $f_i=\cos(dist(e_i, x))$ as the set of first eigenfunctions for $e_i=(0, \cdots, 1, \cdots, 0)\in \mathbb R^{n+1}$. i.e. $\Delta f_i=\lambda_1 f$, where $\lambda_1$ is ...
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Has the notion of "space" been reconsidered in 20th century?
The original title, "has the bases of geometry been reconsidered in 20th century" of this question refers to Riemann's paper "On the Hypotheses which lie at the Bases of Geometry"， an English version ...
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About Thom Theorem (representation submanifold for $H_{n-2}(M^n)$)
Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold.
And in the Harper and ...
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Berger's theorem on Riemannian holonomy applied to the orthogonal frame bundle.
Let $M$ be a compact Riemannian manifold and $TM$ be its tangent bundle. Given a initial point-vector $(x,v) \in TM$ and a curve $\alpha:[0,1] \to M$ starting at $x$ we can parallel transport $(x,v)$ ...
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Do transvers foliations induce complex structure?
Hallo,
I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the ...
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Green functions on Riemann surfaces
Let $(M,g)$ be a compact Rieamnnian surface without boundary and $\Delta_g$ be the Lapalce operator. We note $\lambda_i$ and $\phi_i$ the eigenvalues and eigenunctions of $\Delta_g$. Let also $G_g$ ...
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Curvature as metric invariant
This is quite well-known: the ONLY metric invariants are curvature, its higher
derivatives, and any possible contractions between them.
The meaning of an invariant is, to put it simply, a tensor ...
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Differentiation of Logarithm Map in Riemannian Geometry
I have a simple question regarding the differentiation of the logarithm mapping in Riemannian manifolds:
Assume that $M$ is a compact Riemannian manifold, isometrically embedded into $\mathbb{R}^n$.
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Holonomy of a Kähler manifold
Hi,
I have the following question: Let $(M,J, \omega)$ be a Kähler manifold (not necessary compact). We know that the holonomy group is a subgroup of $U_{n}$. Let $\Omega$ be a constant ($\nabla \...
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Fundamental groups of compact manifolds with non-negative Ricci curvature.
I would like to find an appropriate reference for the following statement:
Statement. Let $M$ be a compact Riemannian manifold with non-negative Ricci curvature.
Then $\pi_1(M)$ is virtually abelian.
...
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Perelman's example on nonuniqueness of tangent cones at infinity
Perelman has an example on manifolds with nonunique tangent cones at infinity. The paper is here. It is a complete manifold with positive Ricci curvature, Euclidean volume growth, and quadratic ...
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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 $\...
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Open problems about CMC hypersurfaces with symmetries?
Recently, Andrews and Li announced a complete classification of CMC ($H=const.$) tori in $S^3$, confirming a conjecture of Pinkall and Sterling. Their main result is that any such torus is ...
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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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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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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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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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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 complete)...
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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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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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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^...
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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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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 of $\...
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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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Geometric Mean Value Property
Does anyone know where I could find a proof of a variant of a version of the mean-value property for harmonic functions in Riemannian manifolds? I'm actually more interested in using an elliptic ...
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Homometric $\Rightarrow$ isometric?
Suppose you know that there is a mapping between
two Riemmanian manifolds $M_1$ and $M_2$ such that,
for each $x_1 \in M_1$, the (codimension-1) measure of the set of points
at distance $d$ from $x_1$ ...
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