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.
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Geodesics on zero-curvature regions of closed surfaces of genus > 1 of non-positive curvature
Let $M$ be a 2-dimensional Riemannian manifold of non-positive curvature everywhere, of genus > 1. Let $\textbf{D} \subset \textbf{C}$ be the open unit disc in the complex plane, the universal cover ...
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Convexity and Strong convexity of subsets of Surfaces
In the book Riemannian geometry - modern introduction by Isaac Chavel, three different definitions of convexity are introduced. I am looking for an example of a set which is convex but not strongly ...
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altering curvature on a tessellation representation of a compact surface
I have been reading about tessellation representations of compact surfaces, such as how the square tiling the plane represents the torus. For surfaces of genus > 1 (the ones that interest me), we ...
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Injectivity radius and the cut locus
Consider a connected, complete and compact Riemannian manifold $M$. Is it correct that the following equality holds: $\text{inj}(x)=\text{dist}\left(x,\text{CuL}(x)\right)$? Or in words that the ...
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Are the eigenvalues of the Laplacian of a generic Kähler metric simple?
It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...
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projection of the co-derivative == co-derivative of the projection ?
Hey,
here is the formal question.
M is a riemannian sub-manifold in N. a,b are vector fields such that for each p$\in$M, $a_p$,$b_p$ in $T_p$M $\subset$ $T_p$N
prove
$\nabla^M_b$a = pr($\nabla^...
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Finding covariant derivative of a riemanian submanifold
Hi,
I have a question about properties which are common to a manifold and its submanifolds. I start with the metric.
$ M \subset N, dim(M) = m, dim (N) = m+1 $
let $ g^N $ be the metric of N, so that $...
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Contracting a geodesic on a space of curvature less than 1
I would like to ask for a reference to the following statement (hopefully correct):
Let $M$ be a manifold of sectional curvature at most $1$ and let $\gamma$ be a closed geodesic.
Suppose that $\...
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"Synthetic" proof of geodesic flow equation?
First, let me explain what I mean by "synthetic" in the title, which is a proof that reasons purely axiomatically and does not explicitly invoke local coordinate charts (either via concrete expansions ...
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What are the computationally useful ways of thinking about Killing fields?
One definition of the Killing field is as those vector fields along which the Lie Derivative of the metric vanishes. But for very many calculation purposes the useful way to think of them when dealing ...
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The orthogonal group of a riemannian metric
Let the inner product of the vectors X and Y on a given four dimensional manifold (EDIT: make this R4) be defined as (X*Y) = gikXiYk; using the summation convention for repeated indicies.
Let A be a ...
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Global description of the Levi-Civita connection
I'm interested in finding a global (coordinate-free) description of the Levi-Civita connection on a (possibly infinite-dimensional) Riemannian manifold X.
I'm not looking for a description of this ...
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Frobenius Theorem
Say a manifold M has 3 vector fields S,T and R whose Lie brackets satisfy the equations $[S,T]=R$, $[R,S]=T$ and $[T,R]=S$
Then I suppose the following properties hold for M,
There exists a metric ...
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Harmonic maps in the cotangent bundle
$M$ is a Riemannian manifold with metric $g$ and we have a map $F: M \to T^{\*}M$ with $F(p)=(p,f(p))$ with a 1-form $f$. On $T^{*}M$ we use the Sasaki-metric.
How can I prove or it is wrong?:
$F$ ...
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Surfaces all of whose geodesics are both closed and simple
The Zoll surfaces have the property that all of their geodesics are closed.
If one futher stipulates that all geodesics are also simple, i.e., non-self-intersecting,
does this leave only the sphere?
...
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Definition of the curvature tensor
I came across this following way of defining connection and curvature which is not so obviously equivalent to the definitions as familiar in Riemannian Geometry books like say by Jost.
If $E$ is a ...
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Schwarz Lemma in terms of conformal surfaces or holomorphic curves?
Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting.
Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ...
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Diameter of a circle in an embedded Riemannian manifold
This question was inspired by an answer to the "Magic trick based on deep mathematics" question. I wanted to post it as a comment, but I ran out of characters! I'm sure there must be a collection of ...
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Constant curvature manifolds
In two different books I found these two related statements.
The book by Jost defines a ``locally symmetric space" as one for which the curvature tensor is constant and which is geodesically complete....
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Flat Riemanniann manifolds
Are there Riemanniann manifolds with zero curvature other than open subsets of $\mathbb{R}^n \times \mathbb{T}^m$, where $\mathbb{T}^m$ is an $m$ dimensional torus and $m,n\geq 0$ ?
Does taking ...
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Special Holonomy Groups for Lorentzian Manifolds
Let $X$ be a Riemannian manifold. If $X$ is simply connected, irreducible, and not a symmetric space then we know that the possible holonomy groups of the metric on $X$ are:
1) $O(n)$ General ...
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When is a Riemannian manifold an open subset of a complete one?
Under what conditions can a Riemannian manifold be embedded isometrically as a submanifold of a complete one of the same dimension? There should some kinds of necessary conditions. For instance, ...
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Equivalent singular chains and differential forms, as functionals on forms, on compact Riemannian manifolds
On a compact Riemannian oriented manifold $M$,for each singular $k$-chain $\sigma$ (with real coefficients), $\sigma$ induces a linear functional on the $\mathbb{R}$-vector space of differential k-...
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Existence of Fermi coordinates on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold, $p$ a point on the manifold and $v \in T_p M$. Let $\gamma$ be the geodesic starting at $p$ in the direction $v$. There exists a time $t_f$ such that there ...
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Orthogonal complements in Hilbert bundles
It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle.
What is known about the ...
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Changing coordinates so that one Riemannian metric matches another, up to second derivatives
Let $g$ and $g'$ be two $C^2$-smooth Riemannian metrics defined on neighborhoods $U$ and $U'$ of $0$ in $\mathbb R^2$, respectively. Suppose furthermore that the scalar curvature at the origin is $K$ ...
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Are all Riemannian metrics induced by Euclidean metrics? [Nash Embedding Theorem]
Let $M$ be a smooth manifold. We can get a Riemannian metric on $M$ by at least two methods: first by partitions of unity and second by the Whitney embedding theorem: we can embed $M$ into a ...
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Structure constants to Christoffel Symbols
What is meant when one says that one has chosen a basis of fields on the manifold with ``anolonomy"?
I get the feeling that it is a choice of basis with non-trivial structure constants say $C^{k}_{ij}...
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Killing fields on homogeneous spaces
Let $G$ be a compact lie group and $H$ a closed subgroup and hence think of $G/H$ as a homogeneous space.
Then how are the Killing fields on $G/H$ the projection of the right-invariant vector fields ...
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Tetrad postulate: Implies or results from the metricity of the connection?
Hi,
I see that the tetrad postulate:
$\nabla_{\mu}e_{\nu}^{I}=\partial_{\mu}e_{\nu}^{I}-\Gamma_{\mu\nu}^{\rho}e_{\rho}^{I}+\omega_{\mu J}^{I}e_{\nu}^{J}=0$
Can be merely derived from writing a ...
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Monopole classes on hyperbolic 3-manifolds
Let $M$ be a closed hyperbolic $3$-manifold, and $e \in H^2(M)$ an integral cohomology class which is the first Chern class of a $Spin^c$ structure on $M$. Suppose there is a solution to the monopole ...
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Lower bound on volume of minimal hypersurface contained in a unit ball with curvature bounds
I was just wondering, if I have a geodesic ball of radius one in a manifold M whose sectional curvature lies between -epsilon and epsilon for epsilon small, and the injectivity radius of my manifold ...
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