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MULTILINEAR ALGEBRA
free v.s. / basis
tensor product
univ property
uniqueness
indep on order (proof)
dim & basis (proof)
tensor product of n v.s.
algebra over a field
mixed tensors / tensor algebra
dim & basis
bilinear V*x W*
transf formula for tensor comp
comp of product of 2 tensors
symmetric tensors & algebra
alternating tensors aka exterior forms
characterization of alt tensors (proof)
exterior algebra (proof)
det of endomo

DIFF. MANIFOLDS
top man
compatible LC
diff atlas / equiv diff atlas
diff man
exa non-diff atlas
exa all finite dim v.s. over R are diff man
exa every open set define a diff man
exa n-sphere is a diff man
class of man / poincare
2° axiom, open, closed, hausdorff (proof)
real projective space
connected sum / class of 2-man
diff map X->R^m wrt atlas
def indep from atlas selected
diff map X->Y
rank of diff map in P
rank indep on LC
local emb / emb / smooth emb
whitney emb
characterization of diff map
pointed vectors
diff curves
derivation of germs
TpM=D(p) for a general man
transf formula for basis induced by 2 LC
vector field / diff v.f.
lie brackets
pushforward dFp = diff of F at P
diff of a diffeo is a linear isomo
matrix associated to dF
s-cov tensor field / diff t.f.

RIEM. MANIFOLDS
riem. man / riem metric / local rep
isometry / local iso
tang vector / length of diff curve / dM
topology induced by dM
orientable diff man / def are equivalent
Ω(E1…En)=1 (proof)
volume form
exterior derivative
closed & exact k-form
de rham cohomology
DRC is an algebra over R
pullback F*
F* closed (exact) if…
F* on DRC
DRC is an invariant
(C8-)homotophy
C8-homotopic fns have same pullback
h connected comp / connect man (proof)
contractible top space (diff man)
C8-contractible man is connected (proof)
connected, compact man is orientable or not
DRC of n-sphere
conn comp orient man is not contractible
affine connection
coeff of ∇ wrt 2 LC (proof)
covariant derivative
parallel vector field
unique // v.f.
geodesic
torsion of ∇ / t. tensor
T belongs to… (proof)
torsion-free ∇
levi-civita ∇
riem. compatible connection
curvature of ∇ / c. tensor
R belongs to… (proof)
properties of R for L-C
riem. curvature
sectional curvature
schur
sect curv for spanned sigma
remarks

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