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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Sasaki Metric of the Tangent Bundle over the Hyperbolic Plane
This is a reference request on what are surely well known facts.
Let $M$ be a compact hyperbolic surface and $S(M)$ its unit tangent bundle. It follows from facts about Möebius tranformations in the ...
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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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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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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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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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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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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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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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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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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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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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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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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 ...
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Measures on Riemannian manifolds which are not induced by the volume form of some Riemannian metric
Let $M$ be a smooth oriented manifold. Does there exist a smooth measure $m$ on $M$ which is not induced by the volume form of some Riemannian metric $g$ on $M$? I would say that the set of volume ...
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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 that:...
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Existence, uniqueness, and regularity for linear parabolic PDE on a complete Riemannian manifold
Let $M$ be a smooth manifold with a complete Riemannian metric $g$ and $E$ a smooth vector bundle over $M$ with an inner product and compatible connection $\nabla$. Let $K: E \rightarrow E$ be a ...
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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$$
...
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Geodesic circles on riemannian manifolds
Can one always find, in a compact riemannian manifold, a closed geodesic isometric to a usual circle when endowed with the ambient distance ? For instance, in the usual flat torus, the only geodesics ...
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Completeness of the future null infinity in defining a black hole
I am using these lectures by Rodnianksi and Dafermos as the reference for this question.
In third point in the list on the top of page 19 they emphasize the importance of completeness of the future ...
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Algebraic surfaces and their (intrinsic) geometry
Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and ...
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Identity of the Weyl-Tensor
Let $(M^n,g)$ be a Riemannian manifold and let $W$ be its Weyl tensor. For a given ONB, does the identity
$$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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Eigenforms for Laplacian on a non-flat two-torus
Does anyone know an explicit, exact description of the eigenforms of the Laplacian on a non-flat two-torus?
- 108k
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Injectivity radius of the Sasaki metric
Let $(Q,g)$ be a (compact) Riemannian manifold with injectivity radius $\rho>0$. There is a natural metric $\tilde g$ on the tangent bundle $TQ$ which is known as the Sasaki metric and which makes $...
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Pólya's conjecture on the spectra of the Laplacians
Recently I've learned something about the spectra of the Laplacians. Given a bounded domain $\Omega \subset \mathbb{R}^n$ with $\partial \Omega$ smooth, we can consider eigenfunctions of Dirichlet ...
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Local splitting of the tangent bundle with interesting properties
Let $(M,g)$ be a Riemannian manifold and let $U\subset M$ be an open subset. Suppose that the tangent bundle over $U$ splits into two orthogonal distributions $TU=\mathcal{E}\oplus \mathcal{F}$.
Is ...
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intersection of geodesiques
Let $(M,g)$ be a closed riemannian surface . let $\alpha$ be a simple closed geodesique . does there is exist a simple closed geodesic $\beta$ that intersect alpha at only 1 point p such that $[\...
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Good Surface,Bad Surface-Surface classification
Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help.
We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by $...
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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} ...
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Metric associated to a Connection on a Vector Bundle
General question: Given a vector bundle $E \rightarrow M$ on a complex manifold $M$, and a connection $\nabla$ on $E$, is it possible to find an Hermitian structure on $E$ such that $\nabla$ is the ...
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Holonomy group of cotangent bundle
Is the holonomy group of the cotangent bundle, of a compact riemannian manifold, with respect to te standard symplectic structure equal to $SU(n)$, where $n$ is the dimension of the riemannian ...
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Full isometry groups of Stiefel and Grassmann manifolds
Hi,
I'm looking for a reference for the full isometry groups of the
(i) complex Stiefel manifolds $U(m)/U(m-l)$, either for the Euclidean metric (i.e. identifying it with orthonormal $m \times l$-...
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Partitions of Unity
Fix a metric $g$ on a smooth, closed manifold $\mathcal{M}$. Take a finite subcover of the manifold from its atlas. Is it true that any smooth partition of unity subordinate to this cover has ...
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Integration By Parts on Non-compact Manifolds
This is undoubtedly a very easy question, but perhaps there are some subtleties. Under what circumstances can we integrate by parts over a non-compact Riemannian manifold? I am aware that having ...
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Is it overkill to invoke Kirszbraun theorem to prove the following fact ?
Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there ...
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Best metrics on exotic R^4
What is known about the existence of complete metrics with good properties (e.g., Einstein, constant scalar curvature, etc...) on exotic ${\bf R}^4$s? Note, that some exotic ${\bf R}^4$s have non-...
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A bounded homogeneous space which fails to be symmetric?
Do we have examples of a contractible bounded open set $D\subseteq\mathbf{C}^n$ such that $Hol(D)$ (the group of biholomorphisms $f:D\rightarrow D$) acts transitively on $D$ but such that there exists ...
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Fubini Study Metric and Einstein constant
Hi all,
it is well known that the complex projective space with the fubini study metric is Einstein, but what is the explicit value, i.e. for which $\mu$ does $Ric=\mu g$ hold?
Moreover, I would ...
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How the Jacobi metrics may be useful in mechanics with or without constraints?
A mechanical system $(Q,K,V)$ is specified by the configuration space $Q,$ the potential energy $V\in C^\infty(Q),$ and the kinetic energy $K=K_g$ given by a Riemannian metric $g$ on $Q.$
If $V{<}...
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Cheeger's Finiteness Theorem and Lipschitz Constant
Cheeger's Finiteness Theorem states that
For each positive numbers $D,v,n$, the
number of diffeomorphism classes of Riemannian manifolds $M$ with
$Diameter(M)\le D$, $Vol(M)\ge v$, and $|K(M)|\le 1$ ...
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space of geodesics
hallo,
i have the following problem: Let $(M,g)$ be a compact Riemannian manifold with metric $g$ and $\nabla$ be the Levi-Civita Connection. Denote by $G(M) =${$\gamma: \mathbb{R} \rightarrow M | \...
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Which vector bundle are the Christoffel symbols sections of?
The collection of Christoffel symbols $\Gamma_{ij}^k$ of a connection (or of a metric) on a smooth manifold $M$ is not the collection of components of a tensor field in some local chart, i.e. they don'...
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Symmetries vs. Bound in codimension of Nash isometric embedding
Let $(M^m,g)$ be a compact smooth Riemannian manifold of dimension $m$. From the celebrated Nash Embedding Theorem, we know there exists a (smooth) isometric embedding $M\hookrightarrow\mathbb R^n$ on ...
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Riemannian metric on a flag variety
$\def\C{\mathbb{C}}\def\CP{\mathbb{CP}}$Every complex projective space $\CP^n$ has a natural Riemannian metric, the Fubini–Study metric, which is defined via the quotient definition of $\CP^n = \...
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Metric Connections on a Lie Group
A Lie group has three standard Cartan connections; the (-)-connection, the (0)-connection, and the (+)-connection. The (0)-connection is Levi-Civita with the associated metric the bi-invariant metric. ...
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Compactness of solutions to parabolic equations (parabolic regularity)
I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature.
For each $s>0$, I have a ...
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eigenspinors of Dirac operator
$M$ compact manifold. Let $\lambda$ be an eigenvalue for the Dirac operator of multiplicity greater than 2. I'm interested in showing the existence of two linearly independant eigenspinors $u$ and $v$ ...
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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 ...
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Methods for generating metrics and minimizing variational dynamics of particles (masses or charges) on n-dimensional smooth manifolds
I am attempting to investigate transformations between two distinct sets of vertices on n-dimensional manifolds with a minimal change in the fundamental shape of the vertices. I will give some ...
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Large geodesically convex subsets of tori
Let $X=\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and let $E$ be a proper open subset of $X$. We say $E$ is geodesically convex if for any $x,y\in E$ the shortest geodesic connecting $x$ and $y$ lies in $E$....
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Behavior of sectional curvature under metric deformations
Metric deformation:
Let $(M,g_0)$ be a Riemannian manifold and consider a (sufficiently smooth) deformation of $g_0$, $$g_t=g_0+th+O(t^2), \quad 0< t<\varepsilon $$ where $h$ is some symmetric (...
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