All Questions
Tagged with riemannian-geometry smooth-manifolds
171 questions
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Geodesics (Local vs Global)
Let $M$ be a Riemannian manifold, and $p,q\in M$ two points. Now, if $M$ is a complete metric space the Hopf–Rinow theorem ensures that there is a geodesic $C\subset M$ joining $p$ and $q$ that ...
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2
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0
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174
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Special Riemannian metric on the product
Let $M^2$ be an orientable surface with boundary endowed with a Riemannian metric $g$. We know that if we put in the manifold $M \times \mathbb{R}$ the product metric $\overline{g} = g + dt^2$, then ...
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1
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0
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123
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Subdividing a Compact Bounded Curvature Manifold into Charts with Bounded Lipschitz Constant
Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$ cover $P$ of $M$; that is ...
5
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1
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495
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Volume comparison on Grassmannian
Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......
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0
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0
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90
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The idealizer of the space of vector fields with vanishing divergence
The space of smooth vector fields on a manifold is counted as a Lie algebra with the usual Lie bracket structure.
Is there a Riemannian manifold of dimension at least $2$ which satisfies either of ...
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5
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2
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377
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Existence of an isotopy in Riemannian manifold
Let $(M,g)$ be a Riemannian manifold, and $p,q\in M$ be two fixed points. We assume $p,q$ are close enough. Say, we assume $p$ and $q$ are in the same normal coordinate chart. It is clear that there ...
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4
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1
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Are complete minimal submanifolds closed?
Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset?
What about the case in which the ambient manifold is an euclidean space?
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18
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3
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2k
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Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators
EDIT: According to some comments on this post I revise the title to remove the misunderestanding.
Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...
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216
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A geometric rank of Riemannian manifolds
There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks:
The maximum number of global independent vector fields which can be defined ...
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1
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159
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Special spheres: principal curvatures with different signs
For $\varepsilon > 0$, we say that a closed, connected and oriented immersed hypersurface $M^n$ of a riemannian manifold $(N^{n+1},g)$ is $\varepsilon$-convex whenever all principal curvatures of $...
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Smoothness of the closest point on a submanifold
Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold.
Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...
- 3,223
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116
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$\ell_p$ geodesic distance on smooth Riemannian manifold and Logarithmic Sobolev Inequalities
Bear with me, I'm not a professional geometer. Recently, I've been studying Logarithmic Sobolev Inequalities (LSI) for probability distributions on manifolds (e.g as done in works of Bobkvo et al. ...
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1
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423
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Riemannian Manifolds of Bounded Curvature
I am a complete newbie Riemannian Geometry with a particular application in mind so please excuse a lack of rigor in the question.
Suppose I have a manifold with sectional curvature everywhere ...
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1
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0
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54
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Extension of a function from a subset of a manifold to the unit sphere
There is a line in this following paper (page no- 221, the paragraph before the Lemma 2.2)
Otsu, Yukio; Shiohama, Katsuhiro; Yamaguchi, Takao, A new version of differentiable sphere theorem, Invent. ...
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2
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1
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720
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Riemannian manifolds: every compact subset is contained in a connected relatively compact open subset [closed]
While working on some problem (not relevant here), it turned out to be convenient to be able to enclose arbitrary compact subsets in "nicer" compact subsets, hence the question:
if $(M,g)$ is a ...
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-2
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1
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112
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Bochner theorem for complete manifolds
Let $(M,g)$ be a complete oriented Riemannian manifold. If $Ric ≥ 0$ on $M$, then any harmonic 1-form $\alpha$ is parallel i.e $x\to |\alpha|^2_{g,x}$ constant?
- 26.3k
2
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1
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126
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Stokes theorem outside of divisor
Let $X$ be a compact Kähler manifold and $D$ be a snc divisor on it. Then on $X\setminus D$ Stokes theorem holds true
$\int_{X\setminus D}\Delta\alpha=0\; \; \; ?$
In this case $X\setminus D$ is non-...
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0
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128
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Is this possible to calculate norm of a vector using its inner product with another vector
Given a manifold $M$ with two Riemannian metrics $g_1$ and $g_2$ on $M$, a vector field $\xi$, and a smooth function $f:M\to \mathbb{R}$, is this possible to calculate $g_2(\xi,\xi)$ having the ...
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1
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137
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Geodesic-like curves stemming from the heat kernel on a manifold
Consider a smooth $n$-dimensional Riemannian manifold $M$ with sufficiently nice geometric and topological properties such that there exist a unique heat kernel $(t,x,y) \mapsto p(t,x,y)$ on it ($t>...
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124
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How do conformal maps affect curvature?
Let $(\overline{M}^{n+1}, \langle \cdot, \cdot \rangle)$ be a riemannian manifold with riemannian connection $\overline{\nabla}$ and consider $M^n \subset \overline{M}$ an orientable hypersurface with ...
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197
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Regarding a proof in the surgery theorem by Gromov and Lawson
I have a question regarding a proof in the article The classification of simply connected manifolds of positive scalar curvature written by Gromov and Lawson. The precise reference is:
Gromov, ...
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1
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264
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What are the minimal local models for Riemannian manifolds? A local question about isometric embeddings
There are many results about isometric embeddings of Riemannian manifolds but I haven't been able to find one that quite answers this question (which I believe must have some kind answer in the ...
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1
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225
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Extending fibre metrics of submanifolds to Riemannian metrics
Let $M$ be a smooth manifold and $S\subseteq M$ a properly embedded smooth submanifold. Suppose that we have a fibre metric on $TM|_S$, i.e. a positive definite real inner-product on $T_pM$ for all $p\...
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2
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535
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Lie algebra of Gradient vector fields(2)
Motivated by this question, is there a $n$ dimensional Riemannian manfold $M$, $n>1$ such that the space of all gradient vector fields is a Lie algebra under the usual Lie bracket ...
- 4,991
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1
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367
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Is there such a connection on the punctured plane?
Is there a connection on $\mathbb{R}^2 \setminus \{0\}$ for which all operators of parallel transports are in the form $$\begin{pmatrix}a&-b\\b&a \end{pmatrix}$$
but the parallel ...
- 108k
10
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2
answers
767
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Intuition for the Drift Term of the Laplace-Beltrami Operator
In coordinates, the Laplace-Beltrami operator on a Riemannian manifold $(M,g)$ can be written as:
$$
\Delta_g = g^{ij}\partial_{ij} - g^{jk}\Gamma^\ell_{jk}\partial_\ell
$$
The second term:
$$
\mu^\...
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0
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191
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A quantity associated with a Riemannian surface
Assume that $E$ is a Riemannian vector bundle, then its structure group is reduced to $O(n)$. Then the structure group of $E \oplus E$ is reduced to $D(O(n) \oplus O(n)) \subset Sp(2n)$ ...
CommunityBot
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1
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429
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The Lie algebra of gradient vector fields
Assume that $M$ is a differentiable manifold. Is there a Riemannian metric on $M$ such that the space of all gradient vector fields on $M$ would be closed under the Lie bracket?
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5
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1
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444
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heat kernel on closed manifolds - error in Chavel's book?
first of all, I am not sure if this question fits here. I asked this question on math.stackexchange also but didn't get an answer so far.
In Isaac Chavel's book Eigenvalues in Riemannian Geometry, ...
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1
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373
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Are metric isometries smooth at the boundary?
Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).
Suppose we have a map $\phi:M \to N$ which satisfies the following properties:
$$(1) \, \,...
- 3,146
12
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0
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381
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Two ways a manifold can have little symmetry
Let $M$ be a closed connected smooth oriented manifold. The following two properties - that $M$ can either enjoy or not - intuitively both mean that $M$ has very little symmetry:
(a) Every self-map $...
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16
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1
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600
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If all balls around two points are isometric... -- manifold version
This question is a natural follow-up of this other question, asked earlier today by wspin.
Let's say that a metric space $(X,d)$ has two poles if:
there are two distinct points $x$, $y$ such that ...
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If $X$ is a compact smooth Riemannian manifold, why don't we integrate on a fundamental domain in the universal cover? [closed]
Let $X$ be a compact connected Riemannian manifold. The metric gives a local volume form. The universal cover is orientable, and has a precompact subspace locally isometric (with the covering metric) ...
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2
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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 ...
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1
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259
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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 ...
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1
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0
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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 (...
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1
answer
2k
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Hodge de Rham operator and orientability
Let $(M,g)$ be a Riemannian manifold. One can consider the exterior algebra bundle $\Lambda(T^*M)$. The sections of this bundle are differential forms, to be noted by $\Omega^k(M)$. One can consider ...
- 6,813
1
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1
answer
326
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material derivative and relation to Riemannian metric
For each $n$ let $N_t$ be an embedded smooth hypersurface in $\mathbb{R}^n$ of dimension $n-1$. $\{N_t\}_t$ is a family of hypersurface that is evolving with some velocity $V$.
Smooth functions on $N$...
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1
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1
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321
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Soliton equation and non-killing potential vector field
I am searching for a non-Killing vector field $\zeta \in\frak X\rm (M)$ where $(M,g)$ is a Riemannian manifold such that
$$\frac12 \frak L_\zeta \rm g+Ric=\lambda g$$
$$ \frak L_\zeta \rm Ric=\lambda \...
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2
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666
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Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$?
On a compact Kahler manifold, let $g$ be the unique Kahler-Einstein metric with $\mathrm{Ricc}(g)=-g$, proved to exist by Yau and Aubin when the first Chern class $C_1(M)<0$.
Question: Does $g$ ...
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2
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Riemannian metrics preserved by diffeomorphisms
Let $f \neq Id$ be a diffeomorphism (of a smooth manifold $M$) which admits some Riemannain metric on $M$ making it an isometry. How many different metrics are preserved by $f$?
Note that $Met(f)=\{g|...
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1
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434
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Smooth manifolds for which every metric is geodesically convex
Are there non compact smooth manifolds which have the property that every Riemannian metric is geodesically convex?
Note that a manifold for which every Riemannian metric is complete must be compact.
...
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1
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0
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485
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Taylor expansion in Riemannian foliations
Take:
$M$ a Riemannian manifold, ${X_0}\in M$,
$N_{X_0}$ a submanifold of $M$ going through ${X_0}$,
and $Z \in N_{X_0}$ in a neighborhood of ${X_0}$.
At ${X_0} \in N_{X_0}$, we consider the ...
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1
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420
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Nowhere vanishing, normalized vector field with bounded derivatives
It is well-known that any non-compact manifold admits a nowhere vanishing vector field. If we have a Riemannian metric we may pick such a vector field and normalize it so that at every point it has ...
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1
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1
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250
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Global geometry measures for Riemannian manifolds
I'm working on a stochastic algorithm and considering it to apply in case of any curved space (manifolds). But in order to make the algorithm as efficient as possible I want to include in it some ...
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1
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1
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213
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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 ...
CommunityBot
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8
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1
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696
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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$. ...
- 108k
1
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1
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348
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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 $\text{Rc}:\Gamma(S^2_+M)\rightarrow\...
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4
votes
1
answer
797
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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}, \dot{\...
- 5,407
4
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1
answer
265
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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 ...
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