Let $(M,g)$ be a Riemannian manifold and $ h = c.g$ for some $c > 0$ . Then the Levi-Civita connections of $g$ and $h$ are same. From the above deduce the relation between corresponding curvature and sectional curvature.

I am able to solve the first part, i.e the Levi-Civita connections are same. But I am unable to solve the second part, i.e the relation between two curvature. Need some help.

  • $\begingroup$ Do you know the formula for the curvatures? $\endgroup$ – user99914 Nov 21 '15 at 8:13
  • $\begingroup$ Yes I know that.. $\endgroup$ – Saikat Nov 21 '15 at 8:13
  • $\begingroup$ Then it should get the expression, as the curvature are given by $\Gamma_{ij}^k$ and $g^{kl}$, and you know that $\Gamma$ is the same while $h^{kl} = c^{-1} g^{kl}$. $\endgroup$ – user99914 Nov 21 '15 at 8:15
  • 1
    $\begingroup$ Cool, so $R(X, Y, Z)$ is not changed as $\nabla$ is not changed. So you only need to change the inner product $\langle \cdot, \cdot \rangle$. $\endgroup$ – user99914 Nov 21 '15 at 8:40
  • 1
    $\begingroup$ oo...I thought $R(x.y,x,y)$ remains unchanged..I forgot that there is an inner product..Now it's fine.Thank you $\endgroup$ – Saikat Nov 21 '15 at 9:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.