Recent content by brown042

So it was just smoothness of exponential map and ODE fact!. Proof is clear now. So you don't like the word "exponential map" because it is a technical term? I am interested in differential geometry and reading Spivak's book. Sometimes I wonder if it is necessary to study all the materials in the...

Anyone has idea?

Maybe then I should say that A is bounded linear transformation? But still isn't continuity of A obvious by construction?

Let q and q' be sufficiently close points on C^oo manifold M. Then is it true that any C^oo curve c:[a,b]-->M where c(a)=q, c(q)=q' can be represented as c(t)=exp_{q}(u(t)v(t)) where u:[a,b]-->R,v:[a,b]-->TM_{q} and ||v||=1? My question comes from Chapter 9 corollary 16 and 17 of Spivak vol1...

Let f:M-->N be isometric immersion. Is it true that we can find a curve in f(M) which is geodesic in N? Thanks.

I appreciate your comment. But still, why is possibility of defining covariant derivative depends only on the equation (*)?

? It's Friday. So nobody has this book?

Comment I found out that the equation comes from the condition that Riemannian metric tensor being well defined. If you have a Volume 1, look at chap 9 problem23. I suppose we can define covariant derivative if Riemannian metric tensor is well defined. But Spivak is saying that the possibility...

Anyone has Spivak volume 2? I aplogize that I can't write the equation using the regular typeset.

I am reading a Vol2 of geometry book by Spivak. On page 220-221 he says that: "Notice that the possibility of defining covariant derivatives depends only on the equation..." The equation is some equality involving Christoffel Symbol. If anyone has this book, could you explain why what he...