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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9
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1answer
311 views

A careful roadtrip from locally symmetric spaces to algebra

I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning completely)...
5
votes
0answers
58 views

Harmonicity on semisimple groups

I asked this on Math.SE and got no answer, so I'll try my luck here. Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in ...
11
votes
4answers
326 views

Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant

The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame ...
4
votes
1answer
146 views

Compact manifolds locally bi-Lipschitz to Euclidean space

I have a compact manifold $M$, and I am allowed to choose some Riemannian metric on it, exactly which I don't care. But I would love it if I could choose the metric $g$ such that every point has an ...
0
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0answers
37 views

Nonstandard support function for the Busemann function

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold. Assume that $M$ contains a ray $\gamma : [0, \infty) \to \mathbb{R}$. Let $b_\gamma$ be the associated Busemann function, i.e., $$ b_\...
2
votes
0answers
51 views

Harmonic functions in tempered distribution sense

Suppose $g$ is a metric on $\mathbb{R}^3$ and $\Omega \subset\subset \mathbb{R}^3$. We assume that $g$ is euclidean outside $\Omega$. My question concerns solutions to $\triangle_g u =0$ that are say ...
20
votes
2answers
551 views

Is there a smooth manifold which admits only rigid metrics?

Does there exist a (finite dimensional) smooth manifold $M$, such that every Riemannian metric on $M$ has no isometries except the identity? Of course, such a manifold must not admit a diffeomorphism ...
4
votes
1answer
75 views

How isometric action on Riemannian manifold acts on cut locus

Assume that $M$ is a simply connected closed Riemannian manifold with no boundary and nonnegative sectional curvaure Assume that ${\bf Z}_n=(g),\ n\geq 3$ acts on $M$ isometrically. Then if $gx=x$, i....
5
votes
1answer
107 views

Are square tiled surfaces dense in the moduli space of translation surfaces?

I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt. At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is dense....
-1
votes
1answer
70 views

Zariski open set in orthogonal grassmanian [closed]

I am confused about the following question. Consider $\mathbb C^4$ endowed with nondegenerate symmetric bilinear form $J:=\left(\begin{matrix}0&0&0&1\\0&0&1&0\\0&1&0&...
4
votes
1answer
136 views

how to define the injectivity radius of manifolds with boundary?

For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum ...
1
vote
0answers
47 views

Is Fano Kahler surface with reverse orientation also Kahler?

In particular, do Fano Kahler surfaces with reverse orientation admit Kahler-Einstein metrics?
9
votes
1answer
109 views

Harmonic function with injective boundary conditions is an immersion?

Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note ...
17
votes
3answers
472 views

Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces

Let $\{M_i\}$ be a sequence of 2-dimensional orientable closed surfaces of genus $g$ with smooth Riemannian metrics with the Gauss curvature at least $-1$ and diameter at most $D$. By the Gromov ...
3
votes
1answer
92 views

coisotropic action on $TS^{2n+1}$

Let $S^{2n+1}$ be the $m$-dimensional sphere in $\mathbb{C}^{n+1}$. Endow $S^{2n+1}$ with the standard metric. Let $S^1$ act by multiplication on $S^{2n+1}$. Then $S^1$ and the canonical action of $SU(...
2
votes
0answers
64 views

A problem of defining addition in a Quotient space

Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Let the set of all re-parameterizations of curves is $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (...
5
votes
2answers
150 views

Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?

(this question is about a particular aspect of a previous question, which was not duly stressed) Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let $$ \widetilde{M}:=\mathbb{P}T^*M $$ be the $...
6
votes
2answers
237 views

Index of Modified Dirac Operator

Let's say we have an oriented compact 4-d Riemannian spin manifold $(M,g)$. Everybody who's anybody has heard about the index of the Dirac operator $D: S^+\rightarrow S-$; it's the $\hat{A}$-genus, ...
2
votes
1answer
138 views

isometric action on the $n$-sphere

Let $S^n$ be the $n$-sphere. If $n=2k+1$ is odd, then we can identify $S^n$ as a subset of $\mathbb{C}^{k+1}$. We define the $S^1$ action on $S^n$ by multiplication, namely $$ \Psi \colon S^1 \times ...
2
votes
0answers
88 views

Willmore functional

Let $(M^2,g)$ and $(\bar{M},\bar{g})$ be two Riemannian manifolds. Suppose that $\mathcal{W}$ is the Willmore functional on the set of immersion functions from $M^2$ to $\bar{M}$. We know that $(\...
2
votes
1answer
64 views

Characterizing left invariant and right-$O_n$ invariant distances on $GL_n$

Consider the group $GL_n(\mathbb{R})$ with its standard topology. It is not hard to show that there exists Riemannian metrics on it which are left-$GL_n$ and right-$O_n$ invariant. (In fact it's ...
4
votes
1answer
139 views

Distance comparison in submanifold versus in the underlying manifold

Let $(M,g)$ be the (underlying) manifold, $(S,g|)$ be a submanifold. Let $a,b,c \in S$. It's not in general true that $d_M(a,b)\leq d_M(a,c) \implies d_S(a,b)\leq d_S(a,c)$. QUESTION I: The above ...
0
votes
1answer
82 views

When are the minimizing geodesics of a totally geodesic submanifold also minimizing in the underlying manifold? [duplicate]

Also asked here: http://math.stackexchange.com/questions/1725787/when-are-the-minimizing-geodesics-of-a-totally-geodesic-submanifold-also-minimiz A reference on totally geodesic submanifold (TGS): ...
1
vote
1answer
109 views

Riemannian metric on a level set of a smooth function on a manifold

Also asked here: http://math.stackexchange.com/questions/1725491/riemannian-metric-on-a-level-set-of-a-smooth-function-on-a-manifold Let $(M,g)$ be a finite or infinite dimensional Riemannian ...
16
votes
1answer
419 views

Just how close can two manifolds be in the Gromov-Hausdorff distance?

Suppose that we have two compact Riemannian manifolds $(M,g)$ and $(N,h)$. Define the Gromov-Hausdorff distance between them in your favorite way, I'll use the infimum of all $\epsilon$ such that ...
6
votes
2answers
172 views

Criterion for deciding the conformal class of a metric on a complete surface

For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function $...
4
votes
1answer
271 views

On the complexification of a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $...
2
votes
2answers
136 views

Finding a specific Global Smooth Function

Any help with this problem would be appreciated. Thanks Suppose $(M^3,g)$ is a smooth compact Riemannian manifold with smooth boundary and $\gamma$ is a simple smooth orientable curve in $M$. Does ...
3
votes
0answers
64 views

Invariant Lagrangians of a connection and its derivatives: how do they look like?

Let $$ L=L(\Gamma,\partial\Gamma,\ldots,\partial^n\Gamma) $$ be a Lagrangian depending on a linear symmetric connection $\Gamma$ on the tangent space of a manifold $M$ together with its derivatives up ...
2
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0answers
108 views

Generalized Isotropic almost complex structures

Let $(M,g)$ be a Riemannian manifold, $TM$ it's tangent bundle, $\mathcal{H}TM$ be the horizontal sub-space of $TTM$ with respect to $g$, $\mathcal{V}TM$ be the vertical sub-space of $TTM$ and $K$ be ...
0
votes
1answer
124 views

Projection of geodesic is geodesic

Background : If a compact Riemannian manifold $M$ with a no curvature condition has disjoint two submanifolds $N_i$, then the distance between them is attained by some minimizing geodesic $c$. If $c'...
2
votes
1answer
68 views

Some manifold which is not totally geodesic in a compact manifold

(1) If $N^k$ is a submanifold in a compact Riemannian manifold $M^{k+m},\ m\geq 1$ s.t. each $p\in N$ has the following property : There exists independent set $\{ X_i\}_{i=1}^k$ tangent to $T_pN$ s.t....
0
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0answers
46 views

Foliation by Umbilic Surfaces

Suppose $(M,g)$ is a simply connected 3 dimensional Riemannian Manifold which is a foliation by Umbilic surfaces. Can I make the claim that there exists a coordinate system $(x_1,x_2,x_3)$ in which ...
-3
votes
1answer
251 views

Well-known name for a certain connection

Have $X \subset \mathbb{R}^3$ be a smoothly embedded surface. Then we try to define a connection on the tangent bundle $TX$ as follows. The tangent space $T\mathbb{R}^3$ is naturally a trivial $\...
1
vote
1answer
147 views

Applying Cheeger and Colding segment inequality

The question turns out quite long and maybe a bit vague, I apologize in advance for that. I am currently trying to understand Cheeger and Colding proof of the almost splitting theorem. Currently I ...
1
vote
0answers
59 views

Possible directions of saddle connections

Let's consider a Riemann surface $X$ of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. A natural parameter on $X$ is a chart for which $q=dz^2$. A $\theta$-trajectory is a maximal ...
0
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0answers
40 views

trace sobolecv inequality for $q=2(n-1)/(n-2)-\varepsilon$ in half space

Can I do the following inequality, for $ u\in D^{1,2}(R^n_+)$, we have $(\int _{R^{n-1}} |u|^{2(n-1)/(n-2)-\varepsilon}dx')^{\frac{ 1}{2(n-1)/(n-2)-\varepsilon } }\leq C ( \int _{ R^n_+}| \nabla u|^...
1
vote
0answers
57 views

Dimension of tangent space to manifold of cross section slices

Given a function $\Phi:\Omega^{\Phi}\subset \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\Omega^{\phi}\subset \mathbb{R}^2\rightarrow\mathbb{R}$, using a ...
5
votes
0answers
241 views

On the curvature tensor with certain conditions

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\lbrace X_1,...,X_n,Y_1,...,Y_m\rbrace $ be a locally orthonormal frame for $M$($3\leq n,m$). If we suppose the curvature tensor $R$ of $g$ ...
1
vote
1answer
67 views

Existence of left-invariant metric on the cotangentbundle of homogeneous spaces?

Let $N$ be a homogeneous space. Therefore we find a Liegroup $G$ and a isotropy-subgroup $K$ of $G$, such that we can identify $N = G/K$. Then we have a canonical action $l\colon G \times G/K \to G/K$ ...
0
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0answers
52 views

Trace Theorem for $q< 2(n-1)/(n-2)$

Can I get a trace theorem inequalite for $R^n_+$: For $q\in [2,2(n-1)/(n-2]$, we have $(\int_ {R^{n-1} } |u|^q dx) ^{2/q}\leq C(\int_{R^n_+} |\nabla u |^2dx)^{1/2}.$
1
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0answers
63 views

Scattering in (pseudo-)Riemannian spaces

I will ask my question in a broad way, leaving a lot of freedom for answers. Suppose that we have a (pseudo-)Riemannian space $(M,g)$ and we fix some ball-like domain $B \subset M$. Suppose you are ...
5
votes
1answer
203 views

Local differential geometry and invariant theory

Can someone please give me pointers to the literature for local differential differential geometry according to invariant theory in the following sense, provided such a literature exists? Start with ...
1
vote
1answer
185 views

Can a conformal map be turned into an isometry? [closed]

Let $f: (M, g) \to (M, g)$ be a conformal diffeomorphism of the riemannian manifold $(M, g)$, with $$ g(f(p))(Df(p) \cdot v_1, Df(p) \cdot v_2) = \mu^2(p) g(p)(v_1, v_2), \quad \forall p \in M, \, \...
1
vote
0answers
45 views

Convex Hull and Least Area Discs in Riemannian 3-Manifolds

Let $M$ be a complete Riemannian 3-manifold and $\gamma \subset M$ a simple closed curve that bounds a least-area disc $D$ - a disc that minimizes the area among all discs bounded by $\gamma$. Let $...
2
votes
0answers
414 views

On Eigenspace of a Bundle Map which is the horizontal part of a complex structure on $TM$

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\mathcal{H}(TM) ‎‎\subseteq‎‎ TTM$ be the horizontal space associated to the Levi-Civita connection of $g$. ‎L‎‎et $\bar{J} : TTM \longrightarrow ...
3
votes
0answers
86 views

Isometric embedding for manifolds with conical singularities?

Motivation: In the 2+1 dimensional gravity theory, solutions of Einstein equation are locally with constant curvature except at the locus of sources. In this paper the authors investigate solutions ...
3
votes
0answers
71 views

Is there a Riemannian metric on the $2$-ball $B^2$ which has negative sectional curvature everywhere, and such that the boundary is concave?

Is there a Riemannian metric on the $2$-ball $B^2$ which has negative sectional curvature everywhere, and such that the boundary is concave; i.e. the geodesic curvature along the boundary points ...
3
votes
0answers
38 views

Dimension of the space of Jacobi fields along $\gamma$ vanishing at $p$ and $q$ is even?

Let $G$ be a compact Lie group with a bi-invariant metric. Let $p$ be a point, and let $q$ be conjugate to $p$ along a geodesic $\gamma$. Does it necessarily follow that the dimension of the space of ...
1
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0answers
50 views

The injectivity radius of $L^2$ metrics

Suppose M is a compact manifold and Rim(M) the space of all Riemannian metrics. Consider $L^2$ metric $ G_g(h,k) = \int_M tr(g^{-1}hg^{-1}k)vol(g)$ where is a Riemannian metric and $ h,k \in T_gM$ . ...