# Questions tagged [riemannian-metric]

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### Geodesics of manifold and geodesics of its submanifold with induced metric

Suppose we have a riemmanian manifold $(\mathcal{M}, g)$. Let $\mathcal{N}$ be properly embedded submanifold of $\mathcal{M}$ with induced metric $\bar{g}$ from $g$. ($\mathcal{N}$ is not ...
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### Definition of gradient of a function $f$ in Riemannian manifold

I'm reading Semi Riemannian Geometry with applications to relativity by Barret Oneill and I'm trying understand the definition of gradient of a function $f$ in Riemannian Manifold. I know that ...
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### A formula of Riemannian metric and its determinant.

Let $g_{ij}$ be a Riemannian metric of a Riemannian manifold, prove: $\large g^{ij}\frac{\partial g_{ij}}{\partial x^k}=\frac{1}{G}\frac{\partial G}{\partial x^k}, G=det(g_{ij}), g^{ij}=g_{ij}^{-1}$ ...
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### Is every function Lipschitz in some Riemannian metric?

Let $M =\mathbb{R}^{n}$ and let $f : M \to \mathbb{R}$ be a smooth function. Define the Riemannian metric $$\|\delta x\|_{x} = (1+ \|\tfrac{\partial f}{\partial x}(x)\|)\|\delta x\|$$ on $M$ (here ...
I learned that in (semi-)Riemannian geometry an interval $ds^{2}=g_{ab}dx^{a}dx^{b}$ is invariant on coordinate transformations. But there is something I do not get in the concept. Consider the simple ...