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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References for metrics in matrix groups

I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...
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62 views

Dimension of curvature invariants

EDIT: Let $V$ be a Euclidean space and let $O(V)$ denotes its orthogonal group. Let $K(V)\subset Sym^2(\wedge^2(V))$ denote the subspace of curvature tensors, i.e. the subspace of elements satisfying ...
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How to investigate the harmonocity of holomorphic vector fields?

Let $(M,g,J)$ be a Kahler manifold and $\nabla$ be its Levi-Civita connection. We know that $\Delta _gX=||\nabla X||^2X$ is the characterizing equation for harmonic unit vector fields. I dont know ...
2
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2answers
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Converse to Lichnerowicz Vanishing Theorem?

The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies ...
9
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1answer
273 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 ...
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55 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
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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 ...
3
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1answer
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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 ...
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0answers
29 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., $$ ...
2
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0answers
40 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 ...
19
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525 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
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1answer
64 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 $gx=x$, ...
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1answer
98 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 ...
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1answer
63 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 ...
4
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1answer
117 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 ...
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46 views

Is Fano Kahler surface with reverse orientation also Kahler?

In particular, do Fano Kahler surfaces with reverse orientation admit Kahler-Einstein metrics?
9
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1answer
86 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
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3answers
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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
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1answer
82 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 ...
2
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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
146 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 ...
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2answers
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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
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1answer
128 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
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0answers
81 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
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1answer
62 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
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1answer
131 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 ...
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1answer
75 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): ...
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1answer
93 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
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1answer
396 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
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2answers
158 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
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1answer
261 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
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2answers
134 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
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0answers
59 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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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 ...
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1answer
121 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 ...
2
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1answer
64 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 $T_pN$ ...
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0answers
43 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 ...
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1answer
248 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 ...
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1answer
117 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 ...
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0answers
58 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 ...
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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 ...
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0answers
53 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 ...
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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 $g$ ...
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1answer
55 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 $l\colon G \times G/K \to G/K$ ...
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0answers
51 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 $(\int_ {R^{n-1} } |u|^q dx) ^{2/q}\leq C(\int_{R^n_+} |\nabla u |^2dx)^{1/2}.$
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60 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
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1answer
196 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 ...
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1answer
179 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, \, ...
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0answers
44 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
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407 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 ...