Return to Question

I am decently experienced on Sobolev spaces on Euclidean spaces, but I just know basic ideas on Riemannian manifolds and want to understand something on Sobolev spaces on them.

Let $(M,g)$ be smooth Riemannian manifold of dimension $m$ and let $\phi:M \to \mathbb{R}^l$. For $x \in M$ and $v \in \mathbb{S}^{m}$ we write $\frac{d}{dt}\phi(\exp_x(tv))|_{t=0}=D\phi(x)v$ and $\frac{d^2}{dt^2}\phi(\exp_x(tv))|_{t=0}=Hess(\phi)(x)(v,v)$ and observe functionals $\int_M \int_{\mathbb{S}^m} |D\phi(x)v|^2 dv dx$ and $\int_M \int_{\mathbb{S}^m} |Hess(\phi)(x)(v,v)|^2 dv dx$. These functionals should be norms equivalent to $L^2(M)$-norms: $\int_M |D\phi(x)|^2 dx$ and $\int_M |Hess(\phi)(x)|^2 dx$, respectively.

Question. How do these $L^2(M)$-norms come into play in the definition of $\phi \in H^2(M,\mathbb{R}^l)$? More precise, can one define Sobolev norm as $\|\phi\|^2_{H^2(M)}=\|\phi\|^2_{L^2(M)} + \|D\phi\|^2_{L^2(M)}+\|Hess(\phi)\|^2_{L^2(M)}$ and $H^2(M)$ as closure of $C^\infty(M)$ with respect to that norm?

I have seen several different (yet equivalent) ways to define Sobolev spaces on Riemannian manifold, but it is not fully clear to me what kind of weak derivatives one observes.

I am decently experienced on Sobolev spaces on Euclidean spaces, but I just know basic ideas on Riemannian manifolds and want to understand something on Sobolev spaces on them.

Let $(M,g)$ be smooth Riemannian manifold of dimension $m$ and let $\phi:M \to \mathbb{R}^l$. For $x \in M$ and $v \in \mathbb{S}^{m}$ we write $\frac{d}{dt}\phi(\exp_x(tv))|_{t=0}=D\phi(x)v$ and $\frac{d^2}{dt^2}\phi(\exp_x(tv))|_{t=0}=Hess(\phi)(x)(v,v)$ and observe functionals $\int_M \int_{\mathbb{S}^m} |D\phi(x)v|^2 dv dx$ and $\int_M \int_{\mathbb{S}^m} |Hess(\phi)(x)(v,v)|^2 dv dx$. These functionals should be norms equivalent to $L^2(M)$-norms: $\int_M |D\phi(x)|^2 dx$ and $\int_M |Hess(\phi)(x)|^2 dx$, respectively.

Question. How do these $L^2(M)$-norms come into play in the definition of $\phi \in H^2(M,\mathbb{R}^l)$? More precise, can one define Sobolev norm as $\|\phi\|^2_{H^2(M)}=\|\phi\|^2_{L^2(M)} + \|D\phi\|^2_{L^2(M)}+\|Hess(\phi)\|^2_{L^2(M)}$ and $H^2(M)$ as closure of $C^\infty(M)$ with respect to that norm?

I have seen several different (yet equivalent) ways to define Sobolev spaces on Riemannian manifold, but it is not fully clear to me what kind of weak derivatives one observes.

UPDATE: A way to tackle this question is Proposition 1 in our paper.

I am decently experienced on Sobolev spaces on Euclidean spaces, but I just know basic ideas on Riemannian manifolds and want to understand something on Sobolev spaces on them.

Let $(M,g)$ be smooth Riemannian manifold of dimension $m$ and let $\phi:M \to \mathbb{R}^l$. For $x \in M$ and $v \in \mathbb{S}^{m}$ we write $\frac{d}{dt}\phi(\exp_x(tv))|_{t=0}=D\phi(x)v$ and $\frac{d^2}{dt^2}\phi(\exp_x(tv))|_{t=0}=Hess(\phi)(x)(v,v)$ and observe functionals $\int_M \int_{\mathbb{S}^m} |D\phi(x)v|^2 dv dx$ and $\int_M \int_{\mathbb{S}^m} |Hess(\phi)(x)(v,v)|^2 dv dx$. These functionals should be norms equivalent to $L^2(M)$-norms: $\int_M |D\phi(x)|^2 dx$ and $\int_M |Hess(\phi)(x)|^2 dx$, respectively.

Question. How do these $L^2(M)$-norms come into play in the definition of $\phi \in H^2(M,\mathbb{R}^l)$?

I have seen several different (yet equivalent) ways to define Sobolev spaces on Riemannian manifold, but it is not fully clear to me what kind of weak derivatives one observes.

I am decently experienced on Sobolev spaces on Euclidean spaces, but I just know basic ideas on Riemannian manifolds and want to understand something on Sobolev spaces on them.

Let $(M,g)$ be smooth Riemannian manifold of dimension $m$ and let $\phi:M \to \mathbb{R}^l$. For $x \in M$ and $v \in \mathbb{S}^{m}$ we write $\frac{d}{dt}\phi(\exp_x(tv))|_{t=0}=D\phi(x)v$ and $\frac{d^2}{dt^2}\phi(\exp_x(tv))|_{t=0}=Hess(\phi)(x)(v,v)$ and observe functionals $\int_M \int_{\mathbb{S}^m} |D\phi(x)v|^2 dv dx$ and $\int_M \int_{\mathbb{S}^m} |Hess(\phi)(x)(v,v)|^2 dv dx$. These functionals should be norms equivalent to $L^2(M)$-norms: $\int_M |D\phi(x)|^2 dx$ and $\int_M |Hess(\phi)(x)|^2 dx$, respectively.

Question. How do these $L^2(M)$-norms come into play in the definition of $\phi \in H^2(M,\mathbb{R}^l)$? More precise, can one define Sobolev norm as $\|\phi\|^2_{H^2(M)}=\|\phi\|^2_{L^2(M)} + \|D\phi\|^2_{L^2(M)}+\|Hess(\phi)\|^2_{L^2(M)}$ and $H^2(M)$ as closure of $C^\infty(M)$ with respect to that norm?

I have seen several different (yet equivalent) ways to define Sobolev spaces on Riemannian manifold, but it is not fully clear to me what kind of weak derivatives one observes.

I am decently experienced on Sobolev spaces on Euclidean spaces, but I just know basic ideas on Riemannian manifolds and want to understand something on Sobolev spaces on them.

Let $(M,g)$ be smooth Riemannian manifold of dimension $m$ and let $\phi:M \to \mathbb{R}^l$. For $x \in M$ and $v \in \mathbb{S}^{m}$ we write $\frac{d}{dt}\phi(\exp_x(tv))|_{t=0}=D\phi(x)v$ and $\frac{d^2}{dt^2}\phi(\exp_x(tv))|_{t=0}=Hess(\phi)(x)(v,v)$ and observe functionals $\int_M \int_{\mathbb{S}^m} |D\phi(x)v|^2 dv dx$ and $\int_M \int_{\mathbb{S}^m} |Hess(\phi)(x)(v,v)|^2 dv dx$. These functionals should be norms equivalent to $L^2(M)$-norms: $\int_M |D\phi(x)|^2 dx$ and $\int_M |Hess(\phi)(x)|^2 dx$,respectively.

Now come my questions.

How do these $L^2(M)$-norms come into play in definition of $\phi \in H^2(M,\mathbb{R}^l)$? I have seen several different (yet equivalent) ways to define Sobolev spaces on Riemannian manifold, but it is not fully clear to me what kind of weak derivatives one observes.

I am decently experienced on Sobolev spaces on Euclidean spaces, but I just know basic ideas on Riemannian manifolds and want to understand something on Sobolev spaces on them.

Let $(M,g)$ be smooth Riemannian manifold of dimension $m$ and let $\phi:M \to \mathbb{R}^l$. For $x \in M$ and $v \in \mathbb{S}^{m}$ we write $\frac{d}{dt}\phi(\exp_x(tv))|_{t=0}=D\phi(x)v$ and $\frac{d^2}{dt^2}\phi(\exp_x(tv))|_{t=0}=Hess(\phi)(x)(v,v)$ and observe functionals $\int_M \int_{\mathbb{S}^m} |D\phi(x)v|^2 dv dx$ and $\int_M \int_{\mathbb{S}^m} |Hess(\phi)(x)(v,v)|^2 dv dx$. These functionals should be norms equivalent to $L^2(M)$-norms: $\int_M |D\phi(x)|^2 dx$ and $\int_M |Hess(\phi)(x)|^2 dx$, respectively.

Question. How do these $L^2(M)$-norms come into play in the definition of $\phi \in H^2(M,\mathbb{R}^l)$?

I have seen several different (yet equivalent) ways to define Sobolev spaces on Riemannian manifold, but it is not fully clear to me what kind of weak derivatives one observes.