**1**

vote

**0**answers

77 views

### Characterization of the Riemann curvature tensor

Let $(M^n,g)$ be a Riemannian manifold, $a\in M$ be a fixed point. It it well known that there exists a coordinate system near $a$ (e.g. the normal one) such that
$$g_{ij}(x)=\delta_{ij}+O(|x|^2).$$
...

**1**

vote

**1**answer

122 views

### Are there compact Riemannian manifolds whith Q-curvature negative?

Are there known examples of compact Riemannian manifolds with Q-curvature negative?

**0**

votes

**1**answer

62 views

### Obstruction to the existence of global isometries on a constant-curvature Riemannian manifold

Let $M$ be an $m$-dimensional simply connected Riemannian manifold that is not geodesically complete. Suppose $M$ has constant sectional curvature.
Because the curvature is constant, locally $M$ ...

**-1**

votes

**0**answers

52 views

### Intuition for Killing vectors in negative-definite Ricci tensor

Theorem 4.3 from Chapter II of Kobayashi's Transformation Groups in Differential Geometry states that, if $(M,\mathrm{g})$ is a Riemannian manifold with negative definite Ricci tensor, then any ...

**2**

votes

**1**answer

180 views

### Do all surfaces (2d riemanian manifolds) admit constant curvature? [closed]

There seems to be a lot of theorems allowing to prove restricted cases of this (eg. uniformization, classification theorem for compact surfaces) . Intuitively, it seems true, but I've never seen a ...

**5**

votes

**1**answer

261 views

### Geodesics on manifolds with boundary

Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. ...

**5**

votes

**2**answers

247 views

### Alternative proof of Varadhan's formula on Riemann manifolds

Consider Varadhan's famous formula for the kernel of the heat equation on a manifold:
$$ \lim_{t \rightarrow 0} t \log h(t,x,y) = - \frac{d(x,y)^2}{4} .$$
I do not have access to his 1967 two ...

**0**

votes

**1**answer

138 views

### A geometric property of singular matrices

Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function.
What matrices belongs to $S$, precisely?
Let ...

**1**

vote

**1**answer

175 views

### Linearisation of Einstein operator

Let $(M,g)$ be a $(m+1)$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$.
The Ricci curvature can be viewed as a differential operator ...

**4**

votes

**1**answer

165 views

### The heat kernel as an exponential of an integral

In $\mathbb{R}^n$, if $\gamma$ is a line segment between $x_0 = \gamma (0)$ and $x = \gamma (t)$, one has the following formula:
$$\frac {\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, ...

**3**

votes

**0**answers

223 views

### Help in how to estimate d-dimensional volume

Let $\mathbf{e_0,e_1,\ldots,e_n}$ $\in$ $R^d$ denote points of a random Poisson point process in $R^d$. which is centered so that $e_0 =0$.
Considering nearest neighbor distances: for a specific ...

**3**

votes

**0**answers

147 views

### “Parallel translate” of a geodesic in the following sense [closed]

Since I'm lazy, I'm shamelessly referring to the following question:
Derivative of Exponential Map
Given a Riemannian manifold $M$, let $\gamma: (a,b) \to M$ be a geodesic and $E$ a parallel vector ...

**2**

votes

**1**answer

65 views

### Does the green kernel converge as a series of functions?

Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following ...

**0**

votes

**0**answers

46 views

### definition of derivative of a vector field [migrated]

i try to define by myself the notion of differentiation of a vector field on a general manifold.
I know that it is a classical subject and that there exist some answers as Lie derivative of a vector ...

**4**

votes

**0**answers

187 views

### Characterization of kernel of Bianchi operator

Let $M$ be a smooth compact manifold, $\mathcal{S}=\Gamma(\odot^2T^*M)$ the Frechet space of symmetric $2$-covariant tensors, and $\mathcal{M}=\Gamma(\odot^2_+T^*M)$ the Frechet manifold of metrics on ...

**8**

votes

**0**answers

118 views

### Closed geodesics avoiding points in hyperbolic surfaces

Let $\Sigma$ be a closed hyperbolic surface. Is it true that for any finite collection of points $x_1,\ldots,x_n\in\Sigma$ there exists a closed geodesic $\gamma$ containing none of them?
Remark: It ...

**6**

votes

**0**answers

213 views

### Have heat kernels for generalized Laplacians on non-compact manifolds been constructed?

Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be ...

**4**

votes

**1**answer

226 views

### Besse p134 Riemann tensor in dimension 4

Does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds? It is said that
$$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4) $$
because we are in dimension ...

**5**

votes

**2**answers

227 views

### Geodesics and Riemannian submersions

Let $X,Y$ be Riemannian manifolds, and $f\colon X\to Y$ be a Riemannian submersion.
Let $\gamma$ be a geodesic on $X$ starting at a point $x\in X$ and which is orthogonal to the fiber $f^{-1}(f(x))$.
...

**7**

votes

**1**answer

168 views

### A question on the twistor space of a manifold

Let $M$ be either (a) self-dual conformal 4-manifold, or (b) hypercomplex $4n$-manifold.
In either case one can construct the twistor space $Z$ (in the case (b) $Z=\mathbb{C}\mathbb{P}^1\times M$ as a ...

**0**

votes

**0**answers

103 views

### Coordinate charts on converging Riemann surfaces

Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as ...

**11**

votes

**0**answers

102 views

### Canonical Immersion of the Double Torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...

**2**

votes

**0**answers

186 views

### The geometry of the holomorph of a Lie group

Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)
Is Hol$(G)$ always a Lie group?
If the answer is yes our main questions:
1.For a left ...

**2**

votes

**1**answer

89 views

### Parallel transport on a Hadamard manifold

Suppose, $X$ is a Hadamard manifold, i.e., a simply connected manifold of non-positive sectional curvature. Fix a point $w$ in $X$. Consider any three points $x, y, z$ in $X$. Let $\tau_{x, w}$ and ...

**3**

votes

**1**answer

86 views

### Certain construction of the Itô integral on manifolds

Let $M$ be a compact Riemannian manifold and let $X \in \mathfrak{X}(\mathbb{R}\times M)$ be a time-dependent vector field on $M$. I want to construct the Itô integral
$$ I(X) = \int_0^T \langle X(t, ...

**1**

vote

**1**answer

347 views

### Compact riemannian manifolds with boundary that have infinite volume?

I am looking for references in the literature pertaining to (essentially riemannian) metric spaces that are compact of infinite volume, such in the following example. Consider a riemannian metric on ...

**14**

votes

**2**answers

609 views

### Exact Definition of Dirac Operator

Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator ...

**1**

vote

**1**answer

87 views

### Isotopy of positively curved surfaces of revolution in $\mathbb{R}^3$

Consider a surface of revolution of positive curvature. My question is, what are the surfaces (with boundary) in $\mathbb{R}^3$ which are isotopic to the surface of revolution, provided each member of ...

**1**

vote

**2**answers

71 views

### Reference request: minimal (maximal) Lorentzian surfaces in $\mathbb{R}^{1,2}$

Let $R^{1,2}$ be the Minkowski 3-space, I would like to know any references about minimal (maximal) orientable Lorentzian surfaces in $\mathbb{R}^{1,2}$, including examples and maybe general theories, ...

**1**

vote

**1**answer

149 views

### Compact surfaces with boundary of constant negative curvature

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or ...

**4**

votes

**1**answer

129 views

### Commutative spectral triples

The corresponence between compact Hausdorff topological spaces and commutative unital $C^*$-algebras is rather well known: Gelfand Najmark theorem gives perfect correspondence between these ...

**0**

votes

**1**answer

177 views

### Some quantities which definitions are (somehow) similar to the classical Divergence

Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...

**4**

votes

**0**answers

90 views

### Isometries of Compact Semisimple Lie Groups

In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...

**3**

votes

**0**answers

102 views

### Bounded functions dense in Sobolev Spaces

Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly ...

**4**

votes

**2**answers

321 views

### Distance function to a submanifold

Let $M$ be a compact Riemannian manifold and $\Sigma\subset M$ a closed submanifold. Given $x\in M$ we define the distance function to $\Sigma$ by $$d_\Sigma(x):=\inf\{d(x,y):y\in \Sigma\},$$ where ...

**6**

votes

**2**answers

396 views

### Do geodesics in SL2R map to geodesics in the hyperbolic plane?

I am looking for a reference/proof/disproof of the following statement.
Equip the Lie group $SL_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle ...

**0**

votes

**0**answers

100 views

### A consequenc of a Lie group act on a Riemannian manifold by isometry

I am learning differential geometry for using this topic in my research. I am stuck to prove following Result which I got in a article.
Formulation:
Let $ f: [0, 1]\rightarrow \mathbb{R}^2$ be a ...

**2**

votes

**2**answers

111 views

### Dilatation of surface diffeomorphisms

Let $S$ be a higher genus surface, let $f\colon S\to S$ be a diffeomorphism and let $f_*\colon H_1(M)\to H_1(M)$ be the induced homology automorphism.
Define dilatation of $f_*$ as the largest ...

**3**

votes

**1**answer

79 views

### Classification of spherical polygons

I need some sort of classification (up to isometry) of spherical polygons (i.e. polygons in $\mathbb{S}^2$ whose edges are given by geodesics) subject to the interior angles and the perimeter of the ...

**2**

votes

**0**answers

105 views

### Continuous family of constant scalar curvature metrics

The question is as follows: Does there exist an example of a (continuous) family of metrics $g_t$ on a compact manifold such that the following properties hold?
All metrics $g_t$ have constant ...

**2**

votes

**1**answer

103 views

### Is the on-diagonal heat kernel “local” with respect to the metric?

Question
Let $X$ be a manifold, and $\mu_A$, $\mu_B$ two Riemannian metric on it which agree on an open subset $U\subset X$, i.e. $\mu_{A\,|U} = \mu_{B\,|U}$. Let $K_A(t;z,w)$ resp. $K_B(t;z,w)$ be ...

**0**

votes

**2**answers

138 views

### Frobenius condition

Suppose X and Y are two unit length vector fields on a Riemannian manifold which are orthogonal at each point. Is it true that the lie bracket of X, Y belongs to the span of the vector fields at each ...

**0**

votes

**1**answer

94 views

### Complex transport equation

Consider an n dimensional Riemannian manifold with boundary.
Let $\Phi$ be a complex valued smooth function defined in M. Does there exist a NONE VANISHING complex valued function $u$ that solves the ...

**1**

vote

**1**answer

169 views

### Sectional curvature as a Hamiltonian on the Grassmanization of the tangent bundle

Edit: According to the comments to the previous version of this question, I remove my essential errors in the question. I thank the commenters very much.
Let $M$ be a n dimensional manifold. ...

**0**

votes

**0**answers

94 views

### Evolution of local oscillation of scalar curvature under Ricci flow

I apologize in advance if the question will turn out to have an obvious answer but my knowledge of Ricci flow is quite limited. Let $(M,g)$ be a smooth compact Riemannian manifold. I denote by $d_{g}$ ...

**12**

votes

**4**answers

593 views

### Green's operator of elliptic differential operator

Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...

**3**

votes

**1**answer

109 views

### Surfaces with specific types of second fundamental form

Given a three dimensional Riemannian manifold $(M,g)$ and a surface $\Sigma \subset M$ can one categorize surfaces where the second fundamental form of $\Sigma$ is a scalar multiple of the induced ...

**4**

votes

**1**answer

104 views

### New setting for tensor definition

This is the standard definition for a tensor in a smooth manifold (Nakahara, for example):
But while reading Salamon's Riemannian Geometry and Holonomy Groups I have found this rather different ...

**5**

votes

**1**answer

88 views

### metric on ${\bf SPD}_n({\mathbb R})$

The cone ${\bf SPD}_n({\mathbb R})$ of symmetric positive definite matrices is endowed with a nice geometrical structure. The midpoint of the (unique) geodesic between $A$ and $B$ is the so-called ...

**9**

votes

**1**answer

745 views

### Formula for the Perimeter of a spherical triangle?

Consider the ordinary sphere $\mathbb{S}^2\subset \mathbb{R}^3$ and a spherical triangle $T\subset \mathbb{S}^2.$ I'm looking for a formula from which the perimeter $P$ of $T$ is "computable" given ...