Consider the Riemannian manifold $(\mathcal{M}, g)$. We consider a conformal change of the metric. Let $\tilde{g}=\phi g$, where $\phi$ is smooth and positive. Moreover, it satisfies $C_1\leq \|\phi\|_{C^2}\leq C_2$ for some positive constants $C_1$ and $C_2$. The geodesic ball in metric $g$ is denoted as $B_r(p)$. Denote the geodesic ball in the conformal metric $\tilde{g}$ as $\tilde{B}_r(p)$. I am wondering if we have $$ B_{c_1r}(p)\subset \tilde{B}_r(p)\subset B_{c_2r}(p)$$ for some positive constants $c_1$ and $c_2$ in a small neighborhood of $p$ by assuming $c_1$ is small and $c_2$ is a little large. I am also curious about how the conformal change is related to the exponential map.

Thanks a lot in advance for your insights.

Consider a compact smooth Riemannian manifold $(\mathcal{M}, g)$. We consider a conformal change of the metric. Let $\tilde{g}=\phi g$, where $\phi$ is smooth and positive. Moreover, it satisfies $C_1\leq \|\phi\|_{C^2}\leq C_2$ for some positive constants $C_1$ and $C_2$. The geodesic ball in metric $g$ is denoted as $B_r(p)$. Denote the geodesic ball in the conformal metric $\tilde{g}$ as $\tilde{B}_r(p)$. I am wondering if we have $$ B_{c_1r}(p)\subset \tilde{B}_r(p)\subset B_{c_2r}(p)$$ for some positive constants $c_1$ and $c_2$ in a small neighborhood of $p$ by assuming $c_1$ is small and $c_2$ is a little large. I am also curious about how the conformal change is related to the exponential map.

Thanks a lot in advance for your insights.