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Results tagged with riemannian-geometry
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user 36688
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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Quadric functions on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold such that the parallel transport along every simple closed curve is the identity operator. A smooth function $f:M\to \mathbb{R}$ is called a quadric function if …
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Triangles in rigid Riemann surfaces
Edit: We thank Vladimir Matveev for his comment on this post which leeds us to revise the question as follows:
Assume that $M_{g}$ is a compact Riemann surface with constant negative cuvature (That …
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Orthonormal vector fields on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$.
Assume that $X: M \to TM$ is a vector field on $M$.
We say that $X$ is an orthonormal vector field …
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totally geodesic submanifold of Heisenberg group
Let $G= \left\{ \begin{pmatrix} 1&a&c\\0&1&b\\0&0&0 \end{pmatrix} \mid a,b,c\in \mathbb{R} \right\}$ be the Heisenberg group. Is there a compact codimension one submanifold of $G$ which is totally …
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A surface on which all regular curves have nowhere vanishing curvature
Let $S$ be a surface in $\mathbb{R}^{3}$ such that every regular curve $\gamma\subset S$ has nowhere vanishing curvature, that is $\kappa(z)\neq 0$ for all $z\in \gamma$. Does this imply that …
- 356
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Integrability of distributions which are invariant under the isometry group
Let $(M.g)$ be a Riemannian manifold and $D$ is a distribution on $M$. Assume that $D$ is invariant under the action of the isometry group of $M$.
Under which conditions such $D$ …
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Is every vector field a gradient vector field with respect to a pseudo metric?
Edit: According to the comment of Prof. Bryant we revise the question as follows:
Assume that $X$ is a smooth vector field on an open manifold $M$, for exmple $\mathbb{R}^2$. Is there a non degenera …
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An special isometric embedding
Let $M$ be a Riemannian manifold and $\gamma$ be a non closed geodesic.
Is there an isometric embedding of $M$ into some $\mathbb{R}^{n}$ which send $\gamma$ into an straight line?
The second questio …
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287
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Compact surfaces whose Gaussian curvature is a subharmonic function
Is there a complete classification of compact surface $S\subset \mathbb{R}^3$ for which $\Delta \kappa \geq 0$ where $\kappa $ is the Gaussian curvature of $S$.
Does every (compact) $2$ dimensional m …
- 356
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An isocline geodesic characterization of $2$ dimensional flat metrics
Lets we have a Riemannian metric on an open subset of the plane which satisfies the following local property.
Local Property: For every point $x$ and every foliation by geodesics around $x$, …
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Some Sandwich properties for complete Riemannian metrics
I search for some properties $P$ for complete Riemannian manifolds which satisfy a kind of Sandwich property.
More precisely I search for those properties $P$ for complete Riemannia …
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A sufficient condition for isometrically embedding of manifolds in the Euclidean space they ...
Assume that $M$ is a submanifold of $\mathbb{R}^n$ and is equipped with a Riemannian metric such that the parallel transports associated with corresponding LC conection preserve the inner products of …
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The volume of spheres in the Sasaki metric
Let $(M,g)$ be a $n$ dimensional Riemannian manifold. The corresponding Sasaki metric on $TM$ is denoted by $g_s$. This enable us to measure the volume $\mathcal{A}(S_x)$ of each unit sphere $S_x …
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Existence of compact totally geodesic submanifold of codimension $1$
Is there a compact orientable Riemannian manifold which does not have a compact totally geodesic submanifold of codimension $1$?
- 356
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A Converse to the Gauss Bonnet Theorem
Let $S$ be a compact surface in $\mathbb{R}^{3}$ with the gauss normal map $N:S\to \mathbb{S}^{2}$. Assme that $\phi;\mathbb{S}^{2}\to S$ is a diffeomorphism. Put $F=N\circ \phi$ and represent $F:\ma …
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