# Tagged Questions

A branch of differential geometry dealing with Riemannian manifolds. *Riemannian manifolds* are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag ...

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### Laplacian of a function restricted to a submanifold

Let $f$ be a sufficiently smooth function on a manifold $S$. Let $M$ be a sub manifold of $S$. Can someone show how I can write $\Delta^M f$ in terms of $\Delta^{S}f$ ? ((i.e the relation between ...
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### Laplacian of a submanifold in an Euclidean space

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$ ($n<m$). Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. ...
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### give a metric on $M$ which in not compatible with given connection?

suppose a connection is given on manifold. question is define a metric on $M$ which in not compatible with given connection ? I have now idea how to define such metric . this Problem is related to ...
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### Diffeomorphism maps geodesics to geodesics

Let $f:M \to N$ a diffeomorphism between riemannian manifolds of the same dimension. What are sufficient conditions for $f$ to map geodesics to geodesics? Of course, if $f$ is an isometry this occurs, ...
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### Formula of a two form for a parallelizable manifold

Let $M^n$ be a parallelizable manifold with the nowhere dependent vector fields $X_1,\ldots, X_n$ forming a basis for the tangent space at each point of $M$. The Lie brackets of these fields are ...
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### Newton-Raphson method on manifolds

Has anyone explored the notion of the Newton-Raphson method on manifolds? Or to put it another way, on $\mathbb R^n$, is there a natural coordinate free way of defining an iterate of the Newton-...
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### Calculate the Euler-Poincaré characteristic of followin surfaces.

Calculate the Euler-Poincaré characteristic of: An ellipsoid. The surfase $S=\left\{ \left(x,y,z\right)\in\mathbb{R}^{3}:x^{2}+y^{10}+z^{6}=1\right\}$. Note: Not how to do this problem, I not ...
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### What value of $c$ makes this Riemannian metric complete?

I was given the following question in my differential geometry class. The instructor does not use a textbook, and gives only theorems and proofs with no examples, so I don't know how to do ...
Show that the convex neighborhood in a Riemannian Manifold are subset contractibles (to any of their points). A subset $A$ of the differential manifold $M$ is contractible to the point $a\in A$ when,...